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IvIBRARY 

OF  THE 

University  of  California. 
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Received ~7vcan^        >  -cSp-. ...  /^"f^^ 

Accession  No.  ^  9/^..X4'^-   '^^^^  ^"^ • 


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» 


M. 


ADAMS'S    NEW    ARITHMETIC. 

IN    WHICH    THE 

PRINCIPLES   OF   OPERATING  BY  NUMBERS 

AR^ 

AKAX.YTZOAZ.Z.Y  ZSXPX.AXNSD, 

AND 

SYNTKETXCALZ.V  APPI.IED; 

THUS 

COMBINING  THE  ADVANTAGES 

TO    BE    DERIVED    BOTH    FROM 

THE  INDUCTIVE  AND  SYNTHETIC  MODE 
OF  INSTRUCTING: 

yHE    WHOLE 

MADE    FAMILIAR  BY  A  GREAT  VARIETY  OF  USEFUL  AND   INTER- 
ESTING   EXAMPLES,  CALCULATED    AT    ONCE  TO   ENGAGE 
THE  PUPIL    IN    THE    STUDY,  AND  TO    GIVE    HIM    A 
-FJ^LL  KNOWLEDGE  OF  FIGURES    IN  THEIR 
j!^l>^ICATION    TO    ALL    THE    PRAC- 
X  TICAL    PURPOSES    OF 
LIFE. 


DESIGNED    FOR    THE    USE    OF 

SCHOOLS   AND   ACADEMIES 

IN  THE  UNITED  STATES. 


BY  DANIEL  ADAMS,  M 


AUTHOR  OP  THE  SCHOLAR'S  ARITHMETIC;  SCHOOL  GEQGRAPHT^  &C, 


ItEENE, 
PUBLISHED  BY  JOHN  PRENTISS 

1828 


DISTRICT  OF  NEW-HAMPSHIRE. 

District  Clerk's  Office. 

Be  it  remembered,  That  on  the  eighteenth  day  of  September,  A.  D.  1827,  in  the 
fifty-second  year  of  the  Independence  of  the  United  States  of  America,  Daniel  Adams, 
of  said  district,  has  deposited  in  this  oflRce  the  title  of  a  book,  the  right  whereof  he  claims 
as  author,  in  the  words  following,  to  wit : 

"  Arithmetic,  in  which  the  Principles  of  operating  by  Numbers  are  analytically  ex  • 
plained,  and  synthetically  applied  :  tlius  combining  the  Advantages  to  be  derived  both 
from  the  inductive  and  synthetic  Mode  of  instructing :  the  whole  made  familiar  by  a 
great  Variety  of  useful  and  interesting  Examples,  calculated  at  once  to  engage  the  Pupil 
in  the  Study,  and  to  give  him  a  full  Knowledge  of  Figures  in  their  Application  to  all  the 
practical  Purposes  of  Life.  Designed  for  tl>e  Use  of  Schools  and  Academies  in  the  United 
States.  By  Dakiel  Adams,  M.  D.  Author  of  the  Scholar's  Arithmetic,  School  Geog- 
raphy, &c." 

In  conformity  to  the  act  of  Congress  of  the  United  States,  entitled, "  An  Act  for 
tho  encouragement  of  learning,  by  securing  the  copies  of  maps,  charts,  and  books, 
to  the  authors  and  proprietors  of  such  copies  during  the  times  therein  mentioned ;" 
and  also  to  an  act,  entitled,  "An  Act  supplementary  to  an  act  for  the  encourage- 
ment of  learning,  by  securing  the  copies  of  maps,  charts,  and  books,  to  the  authors  and 
proprietors  of  such  copies  during  the  times  therein  rnentioned ;  and  extending  the  bene- 
fits thereof  to  the  arts  of  designing,  engraving  and  etching  historical  and  other  prints." 
CHARLES  W.  CUTTER, 

Clerk  of  the  D istrict  of  JVevj-Hampshire. 
A  true  copy. 

Attest,  C.  W.  CUTTER,  Clerk. 


Stereotyped  at  the 


fitereotyped  at  the 
Boston  Type  and  Stereotype  Foundry, 


A3 


There  aro  two  methods  of  teaching, — the  synthetic  and  the  analytic. 
In  the  synthetic  method,  the  pupil  is  first  presented  with  a  general 
view  of  the  science  he  is  studying,  and  afterwards  with  the  particulars 
of  which  it  consists.  The  analytic  method  reverses  this  order:  the 
pupil  is  first  presented  with  the  particulars,  from  which  he  is  led,  by 
certain  natural  and  easy  gradations,  to  those  views  wliich  are  more 
general  and  comprehensive. 

The  Scholar's  Arithmetic,  published  in  1801,  is  synthetic.  If  that 
is  a.  fault  of  the  work,  it  is  a  fault  of  the  times  in  which  it  appeared. 
The  analytic  or  inductive  method  of  teaching,  as  now  applied  to  ele- 
mentary mstruction,  is  among  the  improvements  of  later  years.  Its 
introduction  is  ascribed  to  Pestalozzt,  a  distinguished  teacher  in 
Switzerland.  It  has  been  applied  to  arithmetic,  with  great  ingenuity, 
by  Mr.  Colburn,  in  our  own  country. 

The  analytic  is  unquestionably  the  best  method  of  acquiring  know 
ledge  ;  the  synthetic  is  the  best  metliod  of  recapitulating,  or  remeicing 
it.  In  a  treatise  designed  for  school  education,  both  methods  are  use- 
ful. Such  is  the  plan  of  the  present  undertaking,  v/hich  the  author, 
occupied  as  he  is  with  other  objects  and  pursuits,  would  willingly  have 
forborne,  but  that,  the  demand  for  the  Scholar's  Arithmetic  still  con- 
tinuing, an  obligation,  incurred  by  long-continued  and  extended  pa- 
tronage, did  not  allow  him  to  decline  the  labour  of  a  revisal,  which 
should  adapt  it  to  the  present  more  enlightened  views  of.  teaching  this 
science  in  our  schools.  In  doing  this,  however,  it  has  been  necessary 
to  make  it  a  new  work. 

In  the  execution  of  this  design,  an  analysis  of  each  rule  is  first  given, 
containing  a  familiar  explanation  of  its  various  principles  ;  after- which 
follows  3,  synthesis  of  these  principles,  with  questions  in  form  of  a  sup- 
plement. Nothing  is  taught  dogmatically  /'no  technical  term  is  used 
till  it  has  first  been  defined,  nor  any  principle  inculcated  without  a  pre- 
vious developement  of  its  truth  ;  and  the  pupil  is  made  to  understand 
the  reason  of  each  process  as  he  proceeds. 

The  examples  under  each  rule  are  mostly  of  a  practical  nature,  be- 
ginning with  those  that  are  very  easy,  and  gradually  advancing  to 
those  more  difficult,  till  one  is  introduced  containing  larger  numbers, 
and  which  is  not  easily  solved  in  the  mind  ;  then,  in  a  plain,  familiar 
manner,  the  pupil  is  shown  how  the  solution  may  be  facilitated  by 
figures.  In  this  way  he  is  made  to  see  at  once  their  use  and  their  ap- 
plication. 

At  the  close  of  the  fundamental  rules,  it  has  been  thought  advisable 
to  collect  into  one  clear  view  the  distinguishing  properties  of  those 
rules,  and  to  give  a  number  of  examples  involving  one  or  more  of  them. 
These  exercises  will  prepare  the  pupil  more  readily  to  understand  tho 


4  PREFACE. 

application  of  these  to  the  succeeding  rules ;  and,  besides,  will  serve 
to  interest  him  in  the  science,  since  he  will  find  himseJi*  able,  by  tho 
application  of  a  very  few  principles,  to  solve  many  curious  questions. 

The  arrangement  of  the  subjects  is  that,  which  to  the  author  has 
appeared  most  natural,  and  may  be  seen  by  the  Index.  Fractions  have 
received  all  that  consideration  which  their  importance  demands.  The 
principles  of  a  rule  called  Practice  are  exhibited,  but  its  detail  of  cases 
is  omitted,  as  unnecessary  since  the  adoption  and  general  use  of  federal 
money.  The  Rule  of  Three^  or  Proportion,  is  retained,  and  the  solu- 
tion of  questions  involving  the  principles  of  proportion,  by  analysis^  is 
distinctly  shown. 

The  articles  Alligation,  Arithmetical  and  GeorftBtrical  Progression^ 
Annuities  and  Permutation,  were  prepared  by  Mr.  Ira  Young,  a  mem- 
ber of  Dartmouth  College,  from  whose  knowledge  of  the  subject,  and 
experience  in  teaching,  I  have  derived  important  aid  in  other  parts  of 
the  work. 

The  numerical  paragraphs  are  chiefly  for  the  purpose  of  reference : 
these  references  the  pupil  should  not  be  allowed  to  neglect.  His  at- 
tention also  ought  to  be  particularly  directed,  by  his  instructer,  to  the 
illustration  of  each  particular  principle,  from  which  general  rules  are 
deduced  :  for  this  purpose,  recitations  by  classes  ought  to  be  instituted 
in  every  school  where  arithmetic  is  taught. 

The  supplements  to  the  rules,  and  the  geometrical  demonstrations 
of  the  extraction  of  the  square  and  cube  roots,  are  the  only  traits  of  the 
old  work  preserved  in  the  new. 

DANIEL  ADAMS. 

Mont  Vernon,  (N.  H.)  Sept.  29, 1827. 


i^®a^. 


SIMPLE  NUMBERS.  p^^^ 

Numeraliou  and  Notation, 7 

Addition, 12 

Subtraction,       19 

Multiplication, 26 

Division,        37 

Fractions  arise  from  Division, 42 

Miscellaneous  Questions,  involving-  the  Principles  of  the  preceding  Rules,     *"  62 


COMPOUND  NUMBERS; 


Different  Denominations, 56 

Federal  Money,  - 57 

,  to  find  the  Value  of  Articles  sold  by  the  100,  or  1000,    .     .  64. 

,  Bills*  of  Goods  sold, ..." 68 

Reduction, 69 

TablesofMoney.  Weight,  Measure,  &c 69—82 

Addition  of  Compound  NumlDcrs, 85 

Subtraction,        •     •     • 89 

Multiplication  and  Division, 93 

FRACTIONS. 

Common,  or  Vulgar.    Their  Notation, 101 

Proper,  Improper,  &c 102 

To  change  an  Improper  Fraction  to  a  Whole  or  Mixed  Number,       .     .     .  103 

a  Mixed  Number  to  an  Improper  Fraction, 104 

To  reduce  a  Fraction  to  its  lowest  Terms, .  105 

Greatest  common  Divisor,  how  found, 106 

To  divide  a  Fraction  by  a  Whole  Number  a||wo  ways,        l(M 

To  multiply  a  Fraction  by  a  Whole  Numbe'r;  two  ways, 110 

a  Whole  Number  by  a  Fraction, 112 

one  Fraction  by  another, 113 

General  Rule  for  the  Multiplication  of  Fractions, 114 

To  divide  a  Whole  Number  by  a  Fraction,        1 15 

one  Fraction  by  another, 117 

General  Rule  for  the  Division  of  Fractions, 118 

Addition  and  Subtraction  of  Fractions,        119 

Common  Denominator,  how  found,         120 

Least  Common  Multiple,  how  found, 121 

Rule  for  the  Addition  and  Subtraction  of  Fractions, 124 

Reduction  of  Fractions,        124 

Decimal.    Their  Notation, 132 

Addition  and  Subtraction  of  Decimal  Fractions, 136 

Multiplication  of  Decimal  Fractions, ,  137 

Division  of  Decimal  Fractions, .    .    .  139 

To  reduce  Vulgar  to  Decimal  Fractions, 14ffi 

Reduction  of  Decimal  Fractions,       146 

To  reduce  Shillings,  &c^to  the  Decimal  of  a  Pound,  by  Inspection,     .    .  146 
the  three  first  Decimals  of  a  Pound  to  Shillings,  &c.,  by  Inspectipn,  137^ 


6  INDEXr 

Reduction  of  Currencies; .161 

To  reduce  Endish,  &c.  Currencies  to  Federal  Money,    ...>...  153 

Federal  Money  to  the  Currencies  of  England,  &c 154 

one  Currency  to  the  Par  of  another  Currency, 155 

Interest, 156 

Time,  Rate  per  cent-,  and  Amount  given,  to  find  the  Principal,    ....  164 

Time,  Rate  per  cent.,  and  Interest  given,  to  find  the  Principal,     ....  165 

Principal,  Interest,  and  Time  given,  to  find  the  Rate  per  cent.,    ....  166 

Principal,  Rate  per  cent.,  and  Interest  given,  to  find  the  Time,     ....  167 
To  find  the  Interest  on  Notes,  Bonds,  dtc,  wlien  partial  Payments  have 

been  made, 168 

Compound  Interest, 169 

by  Progression,       229 

Equation  of  Payments, 176 

Ratio,  or  the  Relation  of  Numbers,        177 

Proportion,  or  Single  Rule  of  Three, 179 

Same  Questions,  solved  by  Analysis,  U  65,  ex.  1 — 20. 

Compound  Proportion,  or  Double  Rule  of  Three, 187 

Fellowship,        192 

Taxes,  Method  of  assessing, 195 

Alligation,     .     .     .  - 197 

Duodecimals, 201 

Scale  for  taking  Dimensions  in  Feet  and  Decimals  of  a  Foot,  204 

Involution, 205 1  Evolution, 207 

Extraction  of  the  Square  Root, 207 

Application  and  Use  of  the  Square  Root,  see  Supplement,       .    .    .  212 

Extraction  of  the  Cube  Root, 215 

Application  and  Use  of  the  Cube  Root,  see  Supplement, 


Arithmetical  Progression,     .    .     .    222 
Annuities  at  Compound  Interest,      231 
Practice,  H  29,  ex.  10—19.     H  43. 
Insurance,  IT  82. 
Buying  and  Selling  Stocks,  ![  82. 


Geometrical  Progression,    .    .    .    225 

Permutation, 237 

Commission,  U  82 ;  TI>85,  ex.  5, 6. 
Loss  and  Gain,  TI  82  j  Tl  88,  ex.  1—8. 
Discount,  IT  85,  ex.  6 — 11. 


MISCELLANEOUS  EXAMPLES. 

Barter,  ex.  21—32.  1  Position,  ex.  89—108. 

To  find  the  Area  of  a  Square  or  P^llelogram,  ex.  148—154. 

of  a  Triangle,  ex^lPo — 159. 

Having  the  Diameter  of  a  Circle,  to  find  the  Circuniference  j  or,  having  the 

Circumference,  to  find  the  Diameter,  ex.  171 — 175. 
To  find  the  Area  of  a  Circle,  ex.  176—179. 

of  a  Globe,  ex.  180,  181. 

To  find  the  Solid  Contents  of  a  Globe,  ex.  182—184. 

of  a  Cylinder,  ex.  185—187. 

of  a  Pyraniid,  or  Cone,  ex.  188,  189. 

of  any  Irregular  Body,  ex.  202,  203. 

Gauging,  ex.  190, 191.  |  Mechanical  Powers,  ex.  192—201. 

Forms  of  Notes,  Bonds,  Receipts,  and  Orders, 259 

Book-Keeping,      263 


Ams^mii^ii®< 


NUDXHRATION. 

IT  1.  ^  A  SINGLE  or  individual  thing  is  called  a  wiit^  unity^ 
or  one  ;r  one  and  one  more  are  called  two  ;  two  and  one  more 
are  called  three ;  three  and  one  more  are  called  four  ;  four 
and  one  more  are  called  five  ;  five  and  one  more  are  called 
six;  six  and  one  more  are  called  seven;  seven  and  one  more 
are  called  eight ;  eight  and  one  more  are  called  nine  ;  nine 
and  one  more  are  called  ten^  &c. 

These  terms,  which  are  expressions  for  quantities,  are 
called  numbers,  <  There  are  two  methods  ^of  expressing 
numbers  shorter  than  writing  them  out  in  words ;  one  called 
the  Roman  method  by  letters,*  and  the  other  the  Arabic 
method  by  figures.     The  latter  is  that  in  general  use.^ 

In  the  Arabic  method,  the  nine  first  numbers  have  each 
an  appropriate  character  to  represent  them.     Thus, 

*  In  the  Roman  method  by  letters,!  represents  one;  W,Jive:  X,  ten;  \^, fifty ; 
C,  one  hundred ;  J),  fire  hundred ;  and  M,  one  thousand. 

As  often  as  any  letter  is  repeated,  so  many  times  its  value  is  repeated,  unless  it 
be  a  letter  representing  a  less  number  placed  before  one  representing-  a  greater ; 
then  the  less  number  is  taken  from  the  greater  5  thus,  IV  represents /own^  iX,  nine,  , 
&.C.,  as  will  be  seen  in  the  following 

TABLE. 


One 

I. 

Ninety 

LXXXX.  or  XC. 

Two 

11. 

One  hundred 

C. 

Three 

III. 

Two  hundred 

cc. 

Four 

nil.  or  IV. 

Three  hundred 

ccc. 

Five 

V. 

Four  hundred 

cccc. 

Six 

VI. 

Five  hundred 

D.  or  10  * 

Seven 

VII. 

Six  hundred 

DC. 

Eight 

VIII. 

Seven  hundred 

DCC. 

Nine 

vim.  or  IX. 

Eight  hundred 

DCCC. 

Ten 

X. 

Nine  huixlred 

DCCCC. 

Twenty 

XX. 

One  thousand 

M.  or  ClO.t 

Thirty 

XXX. 

Five  thousand 

lOO.orV.t 

Forty 
Fifty 

XXXX.  or  XL. 

Ten  thousand 

CCIOO.orX. 

L. 

Fifty  thousand 

1003. 

Sixty 

LX. 

Hundred  thousand 

CCCIOOO.orC: 

Seventy 

LXX. 

One  million 

M. 

Eighty 

LXXX. 

Two  million 

MM. 

*  Jo  is  used  instead  of  D  to  represent  five  hundred,  and  ijpr  every  additional  0  au 
Dexed  at  the  right  hand,  the  number  is  increased  ten  times. 

t  CIO  is  used  to  represent  one  thousand,  and  for  every  C  and  0  put  at  each  end,  tb« 
number  is  increased  ten  times. 

X  A  line  over  any  number  increases  its  value  one  thousand  times. 


8 


NUMERATION. 


V  1. 


A  unity  unity y  or  one,  is  represented  by  this  character, 

Two 

Three 

Four         ......... 

Five 

Six  .         .         .         •         . 

Seven       ......... 

Eight 

Nin^e        ......... 

Ten  has  no  appropriate  character  to  represent  it ;  but  is 
considered  as  forming  a  unit  of  a  second  or  higher 
order,  consisting  of  tens,  represented  by  the  same 
character  (1)  as  a  unit  of  the  first  or  lower  order, 
but  is  written  in  the  second  place  from  the  right 
hand,  that  is,  on  the  left  hand  side  of  units  ;  and 
as,  in  this  case,  there  are  no  units  to  be  written 
with  it,  we  write,  in  the  place  of  units,  a  cipher,  (0,) 
which  of  itself  signifies  nothing  ;  thus.       Ten 


One  ten  and  one  unit  are  called 
One  ten  and  two  units  are  called 
One  ten  and  three  units  are  called 
One  ten  and  four  units  are  called 
One  ten  and  ^ve  units  are  called 
One  ten  and  six  units  are  called 
One  ten  and  seven  units  are  called 
One  ten  and  eight  units  are  called 
One  ten  and  nine  units  are  called 
Two  tens  are  called 
Three  tens  are  called 
Four  tens  are  called 
Five  tens  are  called 
Six  tens  are  called 
Seven  tens  are  called 
Eight  tens  are  called 
Nine  tens  are  called 


Eleven 

Twelve 

Thirteen 

Fourteen 

Fifteen 

Sixteen 

Seventeen 

Eighteen 

Nineteen 

Twenty 

Thirty 

Forty 

Fifty 

Sixty 

Seventy 

Eighty 

Ninety 


Ten  tens  are  called  a  hundred^  which  forms  a  unit  of  a 
still  higher  order,  consisting  of  Atmrfrctfe,  represented 
by  the  same  character  (1)  as  a  unit  of  each  of  the 
foregoing  orders,  but  is  written  one  place  further 
toward  the  left  hand,  that  is,  on  the  left  hand  side 
of  tens ;  thus,  .         .         .         .  One  hundred 

One  hundred,  one  ten,  and  one  unit,  are  called 

One  hundred  and  eleven 


1. 
2. 
3. 
4. 
6. 
6. 
7. 
8. 
9. 


10. 
11. 
12. 
13. 
14. 
15 
16, 
17. 
18. 
19. 
20. 
30. 
40. 
50. 
60. 
70. 
80. 
90. 


100. 


lit,. 


IT  2)  3.  NUMERATION.  9 

IT  2«  There  are  three  hundred  sixty-five  days  in  a  year. 
In  this  number  are  contained  all  the  orders  now  described, 
viz.  units,  tens,  and  hundreds.  Let  it  be  recollected,  units 
occupy  the  first  place  on  the  right  hand ;  tens^  the  second 
place  from  the  right  hand;  hundreds^  the  third  place.  This 
number  may  now  be  decomposed,  that  is,  separated  into  parts^ 
exhibiting  each  order  by  itself,  as  follows : — The  highest 
order,  or  hundreds,  are  three,  represented  by  this  character, 
3 ;  but,  that  it  may  be  made  to  occupy  the  third  place,  count- 
ing from  the  right  hand,  it  must  be  followed  by  two  ciphers, 
thus,  300,  (three  hundred.)  The  next  lower  order,  or  tens, 
are  six,  (six  tens  are  sixty,)  represented  by  this  character,  6  ; 
but,  that.it  may  occupy  the  second  place,  which  is  the  place 
of  tens,  it  must  be  followed  by  one  cipher,  thus,  60,  (sixty.) 
The  lowest  order,  or  units,  are  five^,  represented  by  a  single 
character,  thus,  5,  (five.) 

We  may  now  combine  all  these  parts  together,  first  writing 
down  the  five  units  for  the  right  hand  figure,  thus,  5  ;  then 
the  six  tens  (60)  on  the  left  hand  of  the  units,  thus,  65  ;  then 
the  three  hundreds  (300)  on  the  left  hand  of  the  six  tens, 
thus,  365,  which  number,  so  written,  may  be  read  three 
hundred,  six  tens,  and  five  units ;  or,  as  is  more  usual,  three 
hundred  and  sixty-five. 

yi  3.  Hence  it  appears,  that  figures  have  a  different  value 
according  to  the  place  they  occwpry,  counting  from  the  right  hand 
towards  the  left, 

in  Eh  ID 
Take  for  example  the  number  3  3  3,  made  by  the  same 
figure  three  times  repeated.  The  3  on  the  right  hand,  or  in 
the  first  place,  signifies  3  units  ;  the  same  figure,  in  the  second 
place,  signifies  3  tens,  or  thirty ;  its  value  is  now  increased 
ten  times.  Again,  the  same  figure,  in  the  third  place,  signi- 
fies neither  3  units,  nor  3  tens,  but  3  hundreds,  which  is  ten 
times  the  value  of  the  same  figure  in  the  place  immediately 
preceding,  that  is,  in  the  place  of  tens  ;  and  this  is  a  funda- 
mental law  in  notation,Uhat  a  removal  of  one  place  towards 
the  left  increases  the  value  of  a  figure  ten  times.  ^ 

j^Ten  hundred  make  a  thousand,  or  a  unit  of  the  fourth 
order.  Then  follow  tens  and  hundreds  of  thousands,  in  the 
fiame  manner  as  tens  and  hundreds  of  units.     To  thousands 


10  NUMERATION.  IT  S. 

succeed  millions^  billions,  &c.,  to  each  of  which,  as  to  units 
and  to  thousands,  are  appropriated  three  places,*  as  exhi- 
bited in  the  following  examples  : 


O  .  n3 

"^         §         2         s         § 


2  S  V  ^ 


o 


O*  H  pq  S  H  tii 


^^  n3  T3  ^O  '^ 

Qj  q^  OJ  ty  (y 

.^    I    §.^    I    g.^    I    g.5    ^    g.5    I    g.^ 

Example  1st.  3    174592    837463    512 

Example  2d.  3,  1  7  4,  5  9  2,  8  3  7,  4  6  3,  5  1  2, 

**^    rt         *t?  ^  "^  «*-•.**:; 

•TS  O    '^  .    ^  p^  m^        'TIS    'T3  . 

•a.2^    '^.2   «    'S.o    ^    .2.0    g    .2.0    §    -2,0 

Phoj'O    &.<u;:r:    P.aj.2    ^  '^ -^    o^2    Oi^-^ 
&,§  ^  Oh:^         P.3    ^P^S    ^^_S    ^^'S 

ooC^iooH   rttoPQcoo<<   (NoH   f-noM 

To  facilitate  the  reading  of  large  numbers,  it  is  frequently 
practised  to  point  them  off  into  periods  of  three  figures  each^ 
as  in  the  2d  example.  The  names  and  the  order  of  the  pe- 
riods being  known,  this  division  enables  us  to  read  num- 
bers consisting  of  many  figures  as  easily  as  we  can  read 
three  figures  only.  Thus,  the  above  examples  are  read  3 
(three)  Quadrillions,  174  (one  hundred  seventy-four)  Tril- 
lions, 592  (five  hundred  ninety-two)  Billions,  837  (eight 
hundred  thirty-seven)  Millions,  463  (four  hundred  sixty- 
three)  Thousands,  512  (five  hundred  and  twelve.) 

After  the  same  manner  are  read  the  numbers  contained  in 
the  following  r 

*  This  is  according  to  the  French  method  of  counting.  The  English,  after 
hundreds  of  millions,  mstead  of  proceeding  to  billions,  reckon  thousancS,  tens  and 
hundreds  of  thousands  of  millions,  appropriating  six  places,  instead  of  threC;  to 
millious;  billionS;  &c. 


IT  3.  NUMERATION.  11 


NUMERATIOIV  TABLE. 

^  Those  words  at  the  head  of  the 

g  table  are  applicable  to  any  sum  or 

number,  and  must  be  committed  per- 


CO  M 

a  0 


2    ,      ^  "g  fectly  to  memory,  so  as  to  be  readily 

g  g      H  g  applied  on  any. occasion. 

KEhSKHHMEhP  Of  these  characters,  1,  2,  3,  4,  5, 

7  6,  7,  8,  9,  0,  the  nine  first  are  some- 

8  6  times  called  significant   figures,  or 

4  3  2  digits,  in  distinction  from  the  last^ 

7  0  5  4  which,  of  itself,  is  of  no  value,  yet, 

.    .    ..86200  placed  at  the  right  hand  of  another 

.    .    .900371  figure,   it    increases    the    value  of 

.    .   5  0  8  6  0  0  0  that  figure  in  the  same  tenfold  pro- 

.   10302070  portion  as  if  it  had  been  followed  by 

806105409  any  one  of  the  significant  figures. 

Note,  Should  the  pupil  find  any  difficulty  in  reading  the 
following  numbers,  let  him  first  transcribe  them,  and  point 
them  off  into  periods. 

5768  52831209  286297314013 

34120  175264013  5203845761204 

701602  3456720834  13478120673019 

6539285  25037026531  341246801734526 

The  expressing  of  numbers,  (as  now  shown,)  by  figures, 
is  called  Notation,  >  The  reading  of  any  number  set  down  in 
figures,  is  called  Numeration. 

After  being  able  to  read  correctly  all  the  numbers  in  the 
foregoing  table,  the  pupil  may  proceed  to  express  the  fol- 
lowing numbers  by  figures : 

1.  Seventy-six. 

2.  Eight  hundred  and  seven. 

3.  Twelve  hundred,  (that  is,  one  thousand  and  two  hun- 
dred.) 


12  ADDITION   OF    SIMPLE    NUMBERS.  IT  3,  4 

4.  Eighteen  hundred. 

5.  Twenty-seven  hundred  and  ninet^n. 
,6.  Forty-nine  hundred  and  sixty. 

7.  Ninety-two  thousand  and  forty-five. 

8.  One  hundred  thousand. 

9.  Two  millions,  eighty  thousands,  and  seven  hundreds. 
10>  One  hundred  millions,  one  hundred  thousand,  one 

hundred  and  one. 

11.  Fifty-two  millions,  six  thousand,  and  twenty. 

12.  Six  billions,  seven  millions,  eight  thousand,  and  nine 
hundred. 

13.  Ninety-four  billions,  eighteen  thousand,  one  hundred 
and  seventeen. 

14.  One  hundred  thirty-two  billions,  two  hundred  millions, 
and  nine. 

15.  Five  trillions,  sixty  billions,  twelve  millions,  and  ten 
thousand. 

16.  Seven  hundred  trillions,  eighty-six  billions,  and  seven 
millions. 


ADBXTXON 

OF  SIMPLE  NUMBERS. 

IT  4.  1.  James  had  5  peaches,  his  mother  gave  * "'  .  Z 
peaches  more ;  how  many  peaches  had  he  then  ? 

2.  John  bought  a  slate  for  25  cents,  and  a  book  for  eight 
cents  ;  how  many  cents  did  he  give  for  both  ? 

3.  Peter  bought  a  waggon  for  36  cents,  and  sold  it  so  as 
to  gain  9  cents ;  how  many  cents  did  he  get  for  it  ? 

4.  Frank  gave  15  walnuts  to  one  boy,  8  to  another,  and 
had  7  left ;  how  many  walnuts  had  he  at  first  ? 

5.  A  man  bought  a  chaise  for  54  dollars ;  he  expended  8 
dollars  in  repairs,  and  then  sold  it  so  as  to  gain  5  dollars  ; 
how  many  dollars  did  he  get  for  the  chaise  ? 

6.  A  man  bought  3  cows  ;  for  the  first  he  gave  9  dollars, 
for  the  second  he  gave  12  dollars,  and  for  the  other  he  gave 
10  dollars ;  how  many  dollars  did  he  give  for  all  the  cows  ? 

7.  Samuel  bought  an  orange  for  8  cents,  a  book  for  17 
cents,  a  knife  for  20  cents,  and  some  walnuts  for  4  cents  j 
how  many  cents  did  he  spend  ? 


IT  4, 


ADDITION   OF  aiMPLE   NUMBERS. 


13 


8.  A  man  had  3  calves  worth  2  dollars  each,  4  calves 
worth  3  dollars  each,  and  7  calves  worth  5  dollars  each ; 

-^liow  many  calves  had  he  ? 

9.  A  man  sold  a  cow  for  16  dollars,  some  com  for  20  dol- 
lars, wheat  for  25  dollars,  and  butter  for  5  dollars ;  how 
many  dollars  must  he  receive  ? 

The  putting  together  two  or  more  numbers,  (as  iri  the 
foregoing  examples,)  so  as  to  make  one  whole  number^  is 
called  Addition^  and  the  whol^  number  iscalled  the  smuj  or 
amount,- 

10.  One  man  owes  me  5  dollars,  another  owes  me  6 
dollars,  another  8  dollars,  another  14  dollars,  and  another  3 
dollars  ;  what  is  the  amount  due  to  me? 

11.  What  is  the  amount  of  4,  3,  7,  2,  8,  and  9  dollars? 

12.  In  a  certain  school  9  study  grammar,  15  study  arith- 
metic, 20  attend  to  writing,  and  12  study  geography;  what 
is  the  whole  number  of  scholars  ? 

Signs.  (,  A  cross,  -(-,  one  line  horizontal  and  the  other  per- 
pendicular, is  the  sign  of  addition.  It  shows  tliat  numbers, 
with  this  sign  between  them,  are  to  be  added  together.  It 
is  sometimes  read  plus^  which  is  a  Latin  word  signifying 
morft.J 

\  Two  parallel,  horizontal  lines,  -n,  are  the  sign  of  equality.  ; 
It  signifies  that  the  number  before  it  is  equal  to  the  number 
after  it.     Thus,  5  -j-  3  r=  8  is  read  5  and  3  are  8 ;  or,  5  plus 
(that  is,  more)  3  is  equal  to  8. 

In  this  manner  let  the  pupil  be  instructed  ta.commit  the 
following 


ADDITION   TABIiX;. 


2  +  0  = 

2 

2  +  1  = 

3 

24-2=: 

4 

2  +  3  = 

6 

2  +  4  = 

6 

2  +  5  = 

7 

2  +  6  = 

8 

2  +  7  = 

9 

2  +  8  = 

10 

2  +  9  = 

11 

B 

3  +  0  = 

3 

3  +  1  = 

4 

3  +  2  = 

5 

3  +  3  = 

a 

3  +  4  = 

7 

3  +  5  = 

8 

3  +  6  = 

9 

3  +  7^ 

10 

3  +  8  = 

11 

3  +  9  = 

12 

4  +  0=    4 

f  5- 

- 

4+1=    5 

5n 

- 

4  +  2=    6 

6- 

. 

4  +  3=    7 

5- 

« 

4+4=    8 

5- 

-. 

4  +  5=    9 

5- 

« 

4  +  6  =  10 

5- 

- 

4  +  7=11 

5- 

- 

4  +  8=12 

5- 

. 

4  +  9  =  13 

6- 

- 

0=  6 
1=6 
2=  7 
3=  8 
4=9 
5=10 
6  =  11 
7=  Ilk 
8=13 
9  =  14 


14 


ADDITION   OF    SIMPIiE   NUMBERS.  ^4,6 


ADDITION  TABLE— CONTINUED. 

6+0=    6  7  +  0z=    7  8  +  0=    8  9  +  0=    9 

6  +  1=    "7  7+1=    8  8  +  1=    9  9  +  1  =  10 

6  +  2=    8  7+2=    9  8  +  2  =  10  9  +  2  =  11 

6  +  3=    9  7  +  3=10  8  +  3  =  11  9  +  3  =  12 

6  +  4=10  7  +  4=11  8  +  4  =  12  9  +  4  =  13 

6  +  5=11  7  +  5  =  12  8  +  5  =  13  9  +  5=14 

6  +  6=12  7  +  6  =  13  8  +  6  =  14  9  +  6  =  15 

6  +  7  =  13  7  +  7=14  8  +  7  =  15  9  +  7=16 

3  +  8=14  7  +  8=15  8  +  S  =  16  d  +  S^zirf 

6  +  9  =  15  7  +  9=16  8  +  9==  17  9  +  9  =  18 


5  +  9 

z=  how  many  ? 

8  +  7 

=  how  many  ? 

4  +  3 

+  2  =  how  many  ? 

6+4 

+  5  =:  how  many? 

2  +  0 

+  4  +  6  zz:  how  many  ? 

7+1 

+  0  +  8  ~  how  many  ? 

3  +  0 

+  9  +  5  =  how  many  ? 

9  +  2 

+  6  +  4  +  5  rz  how  many  ? 

1+3 

+  5  +  7  +  8  zzz  how  many  ? 

1+2 

+  3+^  +  5  +  6  =  how  many  ? 

8  +  9 

+  0-1-2  +  4  +  5  =  how  many ? 

6  +  2 

+  5  +  0  +  8  +  3  =  how  many  ? 

IT  5.  When  the  numbers  to  be  added  aveismaU^  the  addi- 
tion is  readily  performed  in  the  mind;  but  it  will  frequently 
be  more  convenient,  and  even  necessary,  to  write  the  num- 
bers down  before  adding  them, 

13.  Harry  had  43  cents,  his  father  gave  him  25  cents 
mare ;  how  many  cents  had  he  then  ? 

One  of  these  numbers  contains  4  tens  and  3  units.  The 
other  nimiber  contains  2  tens  and  5  units.  To  unite  these 
two  numbers  together  into  one,  write  them  down  one 
under  the  other,  placing  the  units  of  one  number  directly 
under  umis  of  the  other,  and  the  tens  of  one  number  directly 
under  tens  of  ihe  o4her,  thus : 

43  cejits.  Having  written  the  numbers  in  this  znan* 

25  cents,      ner,  draw  a  line  underneath. 


X  5»  ADDITION    OF    SIMPLE    NUMBERS.  15 

43  cents.  ^^  ^^^^  begin  at  the  right  hand^  and  add 

25  cents.  ^^®  ^  units  of  the   lower  number  to  the  3 

—         *  units  of  the  upper  number,  making  8  units, 

8  which  we  set  down  in  unit's  place. ) 

^   We  then  proceed  to  the  next  column,  and 

43  cents,  add  the  2  tens  of  the  lower  number  to  the 

25  cents.  4  tens  of  the  upper  number,  making  6  tens, 

.      ""       ^       or  60,  which  we  set  down  in  ten's  place, 
^715.  68  cents.     ^^^  ^^^^  ^,^^^1,  -^  ^^^^ 

It  now  appears  t!.at  Harry's  whole  number  of  cents  is  6 
tens  and  8  units,  or  68  cents ;  that  is,  43  -f-  25  ==  68. 

14.  A  farmer  bought  a  chaise  for  210  dollars,  a  horse  for 
70  dollars,  and  a  saddle  for  9  dollars;  what  was  the  whole 
amount  ? 

Write  the  numbers  as  before  directed,  with  units  under 
units,  tens  under  tens,  &c. 

OPERATION. 

Chaise^  210  dollars.  Add  as  before.      The  units   will 

Horsey      70  dollars.  be  9,  the  tens  8,  and  the  hundreds 

Saddle^      9  dollars.  2  ;  that  is,  2 10  -j-  '''0  +  9  =  289. 

Ajiswer,  289  dollars. 

After  the  same  manner  are  performed  the  following  ex- 
amples : 

15.  A  man  had  15  sheep  in  one  pasture,  20  in  another 
pasture,  acd  143  in  another ;  how  mauy  sheep  had  he  in 
the  three  pr,stures  ?     15  -|-  20  +  143  z=z  how  many  ? 

16.  A  man  has  three  farms,  one  containing  500  acres, 
another  213  acres,  and  another  76  acres ;  how  many  acres 
in  the  three  farms  ?     500  -|-  213  +  '''6  =  how  many? 

17.  Bought  a  farm  for  2316  dollars,  and  afterwards  sold 
it  so  as  to  gain  550  dollars;  what  did  I  sell  the  farm  for? 
2316  +  550  =1  how  many  ? 

Hitherto  the  amount  of  any  one  column,  when  added  up, 
has  not  exceeded  9  ;  consequently  has  been  expressed  by  a 
single  figure.  But  it  will  frequently  happen  that  the  amount 
of  a  single  column  will  exceed  9,  requiring /if?o  or  more  figures 
to  express  it. 

18.  There  are  three  bags  of  money.  The  first  contains 
i76  dollars,  the  second,  653  dollars,  the  third,  524  dollarsit 
vhat  is  the  amount  contained  in  all  the  bags  ? 


16  ADDITION    OF    SIMPLE    NUMBERS.  %  6, 

OPERATION.  Writing  down   tlie   numbers  as 

First  bag,       876  already  directed,  we  begin  with  the 

Second  bag,    653  Yight   hand,  or  unit   column,    and 

Third  bag,   ^524  fjj^^j  the  amount  to  be  13,  that  is, 

Amount,       2053  ^^  units  and  1  ten.     Setting  down 

the  3  units,  or  right  hand  figure, 
in  unit's  place,  directly  under  the  column,  wc  reserve  the 
1  ten,  or  left  hand  figure,  to  be  added  with  the  other 
teits,  in  the  next  column,  saying,  1,  which  we  reserved,  to  2 
makes  3,  and  5  are  8,  and  7  are  15,  whi^h  is  5  units  of  its 
0W71  order,  and  1  unit  of  the  next  higher  order,  that  is,  5  tens 
and  1  hundred.  Setting  down  the  5  tens,  or  right  hand  Tjgure, 
directly  under  the  column  of  tens,  we  reserve  the  left  hand 
figure ,  or  1  hundred,  to  be  added  in  the  column  of  hun- 
dreds^ saying,  1  to  5  is  6,  and  6  are  12,  and  8  are  20,  which 
behig  the  last  column,  we  set  down  the  whole  number, 
writing  the  0,  or  right  hand  figure,  directly  under  the  column, 
and  carrying  forward  the  2,  or  left  hand  figure,  to  tlie  next 
place,  or  place  of  thousands.  Wherefore,  we  find  the  whole 
amount  of  money  contained  in  the  three  bags  to  be  2053 
doJlors, — the  answer. 

Proof.  We  may  reverse  the  order,  and,  beginning  at  the 
top,  add  the  figures  downward.  If  the  two  results  are  alike, 
the  work  is  supposed  to  be  right. 

From  the  examples  and  illustrations  now  given,  we  de- 
rive the  following 

RUIiE. 

I.  Write  the  numbers  to  be  added,  one  under  another^ 
placing  units  under  units,  tens  undfer  tens,  &c.,  and  draw  a 
line  underneath. 

II.  Begin  at  the  right  hand  or  unit  column,  and  add  to- 
gether all  the  figures  contained  in  that  column  :  if  the 
amount  does  not  exceed  9,  write  it  under  the  column ;  but 
if  the  amount  exceed  9,  so  that  it  shall  require  two  or  more 
figures  to  express  it,  write  down  the  unit  figure  only  under 
the  column ;  the  figure  or  figures  to  the  left  hand  of  units, 
being  tens,  are  so  many  units  of  the  next  higher  order, 
which,  being  reserved,  must  be  carried  forward,  and  added 
to  the  first  figure  in  the  next  column. 

III.  Add  each  succeeding  column  in  the  same  manner,  aud 
set  down  the  whole  amount  at  the  last  column 


IT  5.  ADDITION    OF    SlJViPLE    NUMBERC.  17 

EXAMPLES    FOR    PRACTICE. 

19.  A  man  bonght  four  loads  of  hay;  one  load  weighed 
1817  pounds,  another  weighed  1950  pounds,  another  2156 
pounds,  and  another  2210  pounds;  what  was  the  amount 
of  hay  purchased  ? 

20.  A  person  owes  A  100  dollars,  B  2160  dollars,  C  785 
dollars  JD  92  dollars  ;  w^hat  is  the  amount  of  his  debts  ? 

21.  A  farmer  raised  in  one  year  1200  bushels  of  wheat, 
850  bushels  of  Indian  corn,  1000  bushels  of  oats,  1086  bush- 
els of  barley,  and  74  bushels  of  pease  ;  what  was  the  whole 
amount  ?  Ans,  4210. 

22.  St.  Paul's  Cathedral,  in  London,  cost  800,000  pounds 
sterling ;  the  Royal  Exchange  80,000  pounds ;  the  Mansion- 
House  40,000  pounds ;  Black  Friars  Bridge  152,840  pounds; 
Westminster  Bridge  389,000  pounds,  and  the  Monument 
13,000  pounds ;  what  is  the  amount  of  these  sums  ? 

Ans,  1,474,840  pounds. 

23.  At  the  census  in  1820,  the  number  of  inhabitants  in 
the  New  England  States  was  as  follows  : — Maine,  298,335 ; 
New  Hampshire,  244,161  ;  Vermont,  235,764;  Massachu- 
setts, 253,287 ;  Rhode  Island,  83,059  ;  Connecticut,  275,248 ; 
what  was  the  whole  number  of  inhabitants,  at  thot  time,  in 
those  States  ?  Ans.  1,389,854. 

24.  From  the  creation  to  the  departure  of  the  Israelites 
from  Egypt  was  2513  years  ;  to  the  siege  of  Troy,  307  years 
more;  to  the  building  of  Solomon's  Temple,  180  years;  to 
the  building  of  Rome,  251  years;  to  the  expulsion  of  the 
kings  from  Rome,  244  years ;  to  the  destruction  oi"  Carthage, 
363  years  ;  to  the  death  of  Julius  Ca^sar^  102  years  ;  to  the 
Christian  era,  44  years  ;  required  the  time  from  the  Crea- 
tion to  the  Christian  era.  Ans.  4004  years. 

25.  26. 

2S63705421061  43675830214(53 

3107429  3  15638  17523497136  2  0 

62530  3  4792  6081275306217 

247135  5652174630128 

8673  8703263472013 


B^ 


18        SUFFLEMiCNT  TO  NUMERATION  ANI)  ADDITION.   tF  6. 

27.  28. 

6364207681023  90237646821S5 

2812345672948  283496732670  8 

6057042087094  9306342167  33  1 

3162835906718  2365478024369 

76  042  86  53  7892  805  06  0  70  80  900 


29.  What  is  the  amount  of  46723,  6742,  and  986  dollars.' 

30,  A  man  has  three  orchards ;  in  the  iirst  there  are  140 
it<^^  that  bear  apples,  and  64  trees  that  bear  peaches  j  in 
ifie  second,  234  trees  bear  apples,  and  73  bear  cherries ;  in 
the  tlurd,  47  trees  bear  plums,  36  bear  pears,  and  25  bear 
cherries ;  how  many  trees  in  all  the  orchards  ? 


rO  NUMERATION  AND  ADDITION. 

QUESTIONS. 

1.  What  is  a  single  or  individual  thing  called?  2.  What 
is  notation  ^  3.  Wliat  are  the  methods  of  notation  now  in 
use  ?  4.  llow  many  are  the  Arabic  characters  or  figures  r 
5.  What  is  numeration  ?  6.  What  is  a  fundamental  law  in 
notation  ?  7.  What  is  addition  ?  8.  What  is  the  rule 
for  addition  ?     9.  What  is  the  result,  or  number  sought. 

called?     10.  W^hat  is  the  sign  of  addition?     11.  of 

equality  ?     12.  How  is  addition  proved  ? 

EXERCISES. 

1.  Washington  was  born  in  the  year  of  our  Lord  1732; 
he  was  67  years  old  when  he  died ;  in  what  year  of  our 
Lord  did  he  die  ? 

2.  The  invasion  of  Greece  by  Xerxes  took  place  481  years 
before  Christ ;  how  long  ago  is  that  this  current  year  1827? 

3.  There  are  two  r.umbers,  the  less  number  is  8671,  the 
difference  between  the  numbers  is  597  j  what  is  the  greater 
number. 


f  By  6*        SUBTRACTION   OF   SIMPLE   NCTMBERS.  1 9 

4.  A  man  borrowed  a  sum  of  money,  and  paid  in  part 
(i84  dollars  ;  the  sum  left  unpaid  was  876  dollars ;  what  was 
the  sum  borrowed  ? 

5.  There  are  four  numbers,  the  first  317,  the  second  812, 
the  third  1350,  and  the  fourth  as  much  as  the  other  three  j 
what  is  the  sum  of  them  all  ? 

6.  A  gentleman  left  his  daughter  16  thousand,  16  hun- 
dred and  16  dollars ;  he  left  his  son  1800  more  than  his 
daughter ;  wha^  was  his  son's  portion,  and  what  was  the 
wnount  of  the  whole  estate  -       a        S  Son's  portion,  19,416. 

^^*'   I  Whole  estate,  37,032. 

7.  A  man,  at  his  death,  left  his  estate  to  his  four  children, 
who,  after  paying  debts  to  the  amount  of  1476  dollars, 
received  4768  dollars  each;  how  much  was  the  whole 
estate?  Ans.  20548. 

8.  A  man  bought  four  hogs,  each  weighing  375  pounds ; 
how  much  did  they  all  weigh?  Am.  1500, 

9.  The  fore  quarters  of  an  ox  weigh  one  hundred  and 
eight  pounds  each,  the  hind  quarters  w^eigh  one  hundred 
and  twenty-four  pounds  each,  the  hide  seventy-six  pounds, 
and  the  tallow  sixty  pounds ;  what  is  tlie  whole  weight  of 
the  ox  ?  Ans,  600 

10.  A  man,  being  asked  his  age,  said  he  was  thirty-four 
years  old  when  his  eldest  son  was  born,  who  was  then  fif* 
teen  years  cf  age ;  what  vv^as  the  age  of  the  father? 

11.  A  man  sold  two  cows  fur  sixteen  dollars  each,  tweiv- 
ty  bushels  of  corn  for  twelve  dollars,  and  one  hundred 
pounds  of  tallow  for  eight  dollars ;  what  was  his  due  ? 


OF    SIMPLE    NUMBERS. 

t  6.  1.  Charles,  having  18  cents,  bought  a  book,  for  which 
he  gave  6  cents ;  how  many  cents  had  he  left  ? 

2.  John  had  12  apples ;  he  gave  5  of  them  to  his  brother  j 
how  many  had  he  left  ? 

3.  Peter  played  at  marbles ;  he  had  23  when  he  began, 
but  when  he  had  done  he  had  only  12 ;  how  many  did  be 
lose? 


20  SUBTRACTION    OF    SIMPLE    NUMBERS.  ^  6. 

4.  A  man  bought  a  cow  for  17  dollars,  and  sold  her  again 
for  22  dollars ;  how  many  dollars  did  he  gain  ? 

^  5.  Charles  is  9  years  old,  and  Andrew  is  13 ;  what  is  the 
difference  in  their  ages  ? 

6.  A  man  borrowed  50  dollars,  and  paid  all  but  18;  how 
many  dollars  did  he  pay?  that  is,  take  18  from  50,  and 
how  many  would  there  be  left  ? 

7.  John  bought  a  book  and  slate  for  33  cents  ;  he  gave  8 
cents  for  the  book ;  what  did  the  slate  cost  him  ? 

8.  Peter  bought  a  waggon  for  36  cents,  and  sold  it  for  45 
cents;  how  many  cents  did  he  gain  by  the  bargain  ? 

9.  Peter  sold  a  waggon  for  45  cents,  which  was  9  cents 
more  than  he  gave  for  it ;  how  many  cents  did  he  give  for 
the  waggon  ? 

10.  A  boy,  being  asked  how  old  he  was,  said  th:it  he  wa* 
25  years  younger  than  his  father,  whose  age  was  33  years ; 
how  old  was  the  boy  ? 

The  taking  of  a  less  number  from  a  greater  (as  in  the 
fbregoing  examples)  is  called  Subtraction.^  The  greater 
^number  is  called  the  jninuend^  the  less  number  the^ mibtrch 
hendj  and  what  is  left  after  subtraction  is  called  the  differ- 
ence, OT^  remainder. 

11.  If  the  minuend  be  8,  and  the  subtrahend  3,  what  is 
the  difference  or  remainder  ?  Ans.  6. 

12.  If  the  subtrahend  be  4,  and  the  minuend  16,  what  is 
the  remainder  ? 

13.  Samuel  bought  a  book  for  20  cents ;  he  paid  down  12 
cents ;  how  many  cents  more  must  he  pay  ? 

Sign.  ^ A  short  horizontal  line,"^ — ,  is  the  sign  of  subtrac- 
tion. It  is  usually  read  minnSy  which  is  a  Latin  word  signi- 
fying less.  It  shows  that  the  number  after  it  is  to  be  taken 
from  the  number  befare  it.  Thus,  8 — 3  r=  5,  is  read  8  mW 
mis  or  less  3  is  equal  to  5 ;  or,  3  from  8  leaves  5.  The 
latter  expression  is  to  be  used  by  the  pupil  in  committing 
tbe  following 


IT  6,  7.        SUBTRACTION    OF    SIMPLE   NU31BERS. 


21 


2  —  2  =  0 

3  —  2  =  1 

4  —  2  =  2 
6  —  2  =  3 

6  —  2  =  4 

7  —  2  =  5 

8  —  2  =  6 

9  —  2  =  7 
10  —  2  =  8 


SUBTRACTION   TABLE. 

6—3  =  3 

7  —  3  =  4 

8  —  3  =  5 

9  —  3  =  6 
10  —  3  =  7 


3  —  3  =  0 

4  —  3=1 

5  —  3  =  2 


4  —  4  =  0 

5  —  4=1 

6  —  4  =  2 

7  —  4  =  3 

8  —  4  =  4 

9  —  4  =  5 
10  —  4  =  6 


5  —  5  =  0 

7  —  7  =  0 

6  —  5=1 

8  —  7  =  1 

7  —  5  =  2 

9  —  7  =  2 

8  —  5  =  3 

10  —  7  =  3 

9—5  =  4 
10  —  5  =  5 

8  — S=iO 

9  —  8  =  1 

6  —  6  =  0 

7  —  6  =  1 

8  —  6  =  2 

9  —  6  =  3 

10  —  8  =  2 

9  —  9  —  0 
10  —  9  =  1 

10—6  =  4 

7  —  3  =:  how  many  ? 

8  —  5  =z  how  many  ? 

9  —  4tzz  how  many  ? 

12  —  3  =:  how  many  ? 

13  —  4  =:  how  many  ? 


18  —  7  zz:  how  many  ? 
28  —  7  =z  how  many  ? 
22  —  13  =  how  many  ? 
33  —  5  =  how  many  ? 
41  —  15  =:  how  many  ? 


IT  7.  When  the  numbers  are  small^  as  in  the  foregoing 
examples,  the  taking  of  a  less  number  from  a  greater  is  rea- 
dily done  in  the  mind;  but  when  the  numbers  are  large^ 
the  operation  is  most  easily  performed  part  at  a  time,  and 
therefore  it  is  necessary  to  write  the  numbers  down  before 
performing  the  g«- ^ration. 

14.  A  farmer,  having  a  flock  of  237  sheep,  lost  114  of 
them  by  disease ;  how  many  had  he  left  ? 

Here  we  have  4  units  to  be  taken  from  7  units,  1  ten  i* 
be  taken  from  3  tens,  and  1  hundred  to  be  taken  from  2 
hundreds.  It  will  therefore  be  most  convenient  to  write  the 
less  number  under  the  greater,  observing,  as  in  addition,  to 
place  units  under  units,  tens  under  tens,  &c.  thus : 

We  now  begin  with  the 


OPERATION. 
From   237  the  minuend^ 
Take   114  the  subtrahend^ 


units,  saying,  4  (units)  from 
7,  (units,)  and  there  remain  3, 
(units,)  which  we  set  down 
directly  under  the  column  in 
unit's  place.  Then,  proceed 
'ng  to  the  next  column,  we  say,  1  (ten)  from  3,  (tens,)  and 
here  remain  2,  (tens,)  which  we  set  down  in  teii^s  place* 


123  the  remainder. 


S2  SUBTRACTION    OF    SIMPLE    NUMBERS.  IH  T 

Proceeding  to  the  next  column,  we  say,  1  (hundred)  from  2, 
(hundreds,)  and  there  remains  1,  (hundred,)  which  we  set 
do^vn  in  hundTeiTs  place,  and  the  work  is  done.  It  now  ap* 
pears,  that  the  number  of  sheep  left  was  123 ;  that  is, 
237—114  =  123. 

After  the  same  manner  are  performed  the  following  ex- 
amples : 

15.  There  are  two  farms;  one  is  valued  at  3750,  and  the 
other  at  1500  dollars ;  what  is  the  difl'erence  in  the  value 
of  the  two  farms  ? 

16.  A  man's  property  is  worth  8560  dollars,  but  he  ha» 
debts  to  the  amount  of  3500  dollars  ;  what  will  remain  aftei 
paying  his  debts  ? 

17.  James,  having  15  cents,  bought  a  pen-knife,  for  which 
he  gave  7  cents;  how  many  cents  had  he  left? 
OPERATION. 

15  cents.  A  difficulty  presents  itself  here  ;  for  we 

7  ceMs*  cannot  take  7  from  5 ;  but  we  can  take  7 

—  ,  ^  from  1 5,  and  there  will  remain  8. 

Scents  left. 

18.  A  man  bought  a  horse  for  85  dollars,  and  a  cow  for 
27  dollars;  what  did  the  horse  cost  him  more  than  the 
cow  ? 

OPERATION.  The  same  difficulty  meets  us  here  as  in 

Horsey  85  the  last  example  ;  we  cannot  take  7  from 
CkfW,  27         5 ;  but  in  the  last  example  the  larger  num- 

—  her  consisted  of  1  ten  ap  ^  5  units,  which 
Difference,  58  together  make  15;  we  therefore  took  7 
from  15.  Here  we  have  8  tens  and  5  units.  We  can  now, 
in  the  mind,  suppose  1  ten  taken  from  the  8  tens,  which 
would  leave  7  tens,  and  this  1  ten  we  can  suppose  joined  to 
the  5  units,  making  16.  We  can  now  take  7  from  15,  as  be- 
fore, and  there  will  remain  8,  which  we  set  down.  The 
taking  of  1  ten  out  of  8  tens,  and  joining  it  with  the  5  units, 
is  called  borrowing  ten.  Proceeding  to  the  next  higher  o> 
der,  or  tens,  we  must  consider  the  upper  figure,  8,  from 
which  we  borrowed,  1  less,  calling  it  7;  then,  taking  2  (tens) 
from  7,  (tens,)  there  will  remain  5,  (tens,)  which  we  set  down, 
making  the  difference  58  dollars.  Or,  instead  of  making 
the  upper  figure  1  less,  calling  it  7,  we  may  make  the  lower 
figure  one  jnore,  calling  it  3,  and  the  result  will  be  thesam&( 
Cor  3  from  8  leaves  5,  the  same  as  2  from  7. 


t  7,  8.        SUBTRACTION   OF    SIMPLE   NUMBERS.  2S 

19.  A  man  borrowed  713  dollars,  and  paid  471  dollars; 
how  many  dollars  did  he  then  owe?  713  —  471=  how 
many  ?  Ans.  242  dollars, 

20.  1612-— 465  z=  how  many  ?  ^7W.  1147. 

21.  43751 — 6782  zz:  how  many  ?  Ans.  36969. 

IT  8.  CThe  pupil  will  readily  perceive,  that  subtraction  is 
the  reverse  of  addition.^ 

22.  A  man  bought  40  sheep,  and  sold  18  of  them;  how 
many  had  he  left  ?  40  —  IS  zz:  how  many  ?     Ans.  22  sheep. 

23.  A  man  sold  18  sheep,  and  had  22  left;  how  many  had 
he  at  first  ?    18  -{-  22  =  how  many  ?  Ans.  40. 

24.  A  man  bought  a  horse  for  75  dollars,  and  a  cow  for 
16  dollars  ;  what  was  the  difference  of  the  costs  ? 

75  —  16  iz:  how  many  ?    Reversed,  59  -f-  16  zz  how  many? 

25.  114  — 103rz:howmany?  Reversed,  11  +  103 zi: how 
many  ? 

26.  143  —  76  zn  how  many  ?  Reversed,  67  +  76  zz:  how 
many  ? 

Hence,  subtraction  may  be  proved  by  addition^  as  in  the 
foregoing  examples,  and-addition  by  subtraction, 

*To  prove  subtraction^  (;^ve  may  add  the  remainder  to  the 
mthtrahend^  and,  if  the  work  is  right,  the  amount  will  be  equal 
to  the  minuend.  "^ 

^Jb  prove  oMition^  we  may  subtract^  successively,  from 
the  amount,  the  several  numbers  which  were  added  to  pro- 
duce it,  and,  if  the  work  is  right,  there  will  be  no  re- 
mainder)  Thus  7  +  8  +  6  zz:  21 ;  proof,  21  —-  6  izr  15,  and 
15  —  8  zz:  7,  and  7  —  7  zz  0. 

From  the  remarks  and  illustrations  now  given,  we  deduce 
tlie  following 

RULE. 

I.  Write  down  the  numbers,  the  less  under  the  greater, 

E lacing  units  under  units,  tens  under  tens,  &c.  and  draw  a 
ne  under  them."\ 

II.  Beginning  with  units,  take  successively  each  figure  in 
the  lower  number  from  the  figure  over  it,  and  write  the  re- 
mainder directly  below. 

IllJ^JWhen  the  figure  in  the  lower  number  exceeds  the 
figure  over  it,  suppose  10  to  be  added  to  the  upper  figure; 
but  in  this  case  we  must  add  1  to  the  lower  figure  in  tha 
Ufizt  column,  before  subtracting.    This  is  called  borrowing  10^^ 


24  SUPPLEMENT    TO    SUBTRACTION.  Tl  8, 

exampl.es  for  practice. 

27.  If  a  farm  and  the  buildings  on  it  be  valued  at,  10000, 
.nd  the  buildings  alone  be  valued  at  4567  dollars,  what  is 

the  value  of  the  land  ? 

28.  The  population  of  New  England,  at  the  census  in 
1800,  was  1,232,454  ;  in  1820  it  was  1,659,854  ;  what  was 
the  increase  in  20  years  ? 

29.  What  is  the  diifereuce  between  7,648,203  and 
928,671  ? 

30.  How  much  must  you  add  to  358,642  to  make 
1,487,945  ? 

31.  A  man  bought  an  estate  for  13,682  dollars,  and  sold  it 
again  for  15,293  dollars  j  did  he  gain  or  lose  by  it?  and  how 
much  ? 

32.  From  364,710,825,193  take  27,940,386,574. 

33.  From  831,025,403,270  take  651,308,604,782. 

34.  From  127,368,047,216,843  take  978,654,827,352. 


TO   SUBTRACTION.  ,' 

QUESTIONS. 

1.  What  is  subtractio7i?     2.  What  is  the  greater  numher 

called  ?     3.  the  less  number  ?     4.  What  is  the  resuk 

or  answer  called  ?  5.  What  is  the  sign  of  subtraction  ? 
6.  What  is  the  rizZe  ?  7.  What  is  understood  by  horrowing 
ten  ?  8.  Of  what  is  subtraction  the  reverse  ?  9.  How  is 
subtraction  proved  ?  10.  How  is  addition  proved  by  sub- 
traction ? 

EXERCISES. 

1.  How  long  from  the  discovery  of  America  by  Colum- 
bus, in  1492,  to  the  commencement  of  the  Revolutionary 
war  in  1775,  which  gained  our  Independence  ? 

2.  Supposing  a  man  to  have  been  born  in  the  year  1773, 
now  old  was  he  in  1827  ? 

3.  Supposing  a  man  to  have  been  80  years  old  in  the  year 
1826,  in  what  year  was  he  born  ? 

4.  There  are  two  numbers,  whose  difference  is  8764 ;  the 
greater  number  is  15687  j  I  demand  the  less  ? 


f  8,         SUPPLEMENT  TO  SUBTRACTION.  2b 

6.  What  number  is  that  which,  taken  from  3794,  leaves 

865  ? 

6.  What  number  is  that  to  which  if  you  add  789,  it  will 
become  6  J50  ? 

7.  In  New  York,  by  the  census  of  1S20,  there  were 
123,706  inhabitants;  in  Boston,  43,940;  liow  many  more 
inhabitants  were  then  in  New  York  than  in  Boston? 

8.  A  man,  possessing  an  estate  of  twelve  thousand  dollars, 
gave  two  thousand  five  hundred  dollars  to  each  of  his  two 
daughters,  and  the  remainder  to  his  son ;  what  was  his  son'» 
share  ? 

9.  From  seventeen  million  take  fifty-six  thousand,  and 
what  will  remain  ? 

10.  VVhat  number,  together  with  these  three,  viz.  1301, 
2561,  and  3120,  will  make  ten  thousand? 

11.  A  man  bought  a  horse  for  one  hundred  and  fourteen 
dollars,  and  a  chaise  for  one  hundred  and  eighty-seven  dol- 
lars ;  how  much  more  did  he  give  for  the  chaise  than  for 
the  horse  ? 

12.  A  man  borrows  7  ten  dollar  bills  and  3  one  dollar 
bills,  and  pays  at  one  time  4  ten  dollar  bills  and  5  one  dol- 
lar bills ;  how  many  ten  dollar  bills  and  one  dollar  bills 
must  he  afterwards  pay  to  cancel  the  debt  ? 

Ans.  2  ten  doll,  bills  and  8  one  d  .11. 

13.  The  greater  of  two  numbers  is  24,  and  the  less  is  IC; 
what  is  their  difference  ? 

14.  The  greater  of  two  numbers  is  24,  and  their  ditTeft- 
cnce  8  ;  w  hat  is  the  less  number  ? 

15.  The  sum  of  two  numbers  is  40,  the  less  is  16  ;  what 
is  the  greater  ? 

16.  A  tree,  p8  feet  high,  was  broken  off  by  the  wind ;  the 
top  part,  which  fell,  was  49  feet  long;  how  hio^b  was  th« 
stump  which  was  left  ? 

17.  Our  pious  ancestors  landed  at  Plymputh,  Massacliti^ 
•elts,  in  1620  ;  how  many  years  since  ? 

18.  A  man  carried  his  produce  to  market ;  he  sold  hit 
pork  for  45  dollars,  his  cheese  for  38  dollars,  and  his  butter 
for  29  dollars;  he  received,  in  pay,  salt  to  ihe  value  of  17 
dollars,  10  dollars  worth  of  sugar,  5  dollars  worth  of  men 
lasses,  and  the  rest  in  money;  how  much  mo!»ey  did  !i« 
receive?  Jus.  MO  (!o liars. 

19.  A  boy  bougfit  a  sled  for  28  cents,  and  gave  14  ceuti 

C 


20  MULTIPLICATION    OF    SIMPLE    NUMBERS.       If  8,  9. 

to  have  it  repaired ;  he  sold  it  for  40  cents ;  did  he  g<iin  or 
lose  by  the  baroaiu  ?  and  liow  much  ? 

20.  One  man  travels  67  miles  in  a  day,  another  man  fol- 
lows at  the  rate  of  42  miles  a  day  ;  if  they  both  start  from 
the  same  place  at  the  same  time,  how  far  v/ill  they  be  apart 

at  the  close  of  the  lirst  day?  of  the  second  r  of 

the  third  ?  of  the  fourth  ? 

21.  One  man  starts  from  Boston  Monday  morning,  and 
travels  at  the  rate  of  40  miles  a  day;  another  starts  from  the 
same  place  Tuesday  morning,  and  follows  on  at  the  rate  of 
7C.  miles  a  day  ;  liow  far  are  they  apart  Tuesday  night  ? 

Alls.   10  miles. 

22.  A  man,  owing  379  dollars,  paid  at  one  time  47  dol- 
lars, at  another  time  84  dollars,  at  another  time  23  dollars, 
and  at  another  time  143  dollars ;  how  much  did  he  then 
owe  ?  Ans.  82  dollars. 

23.  A  man  has  property  to  the  amount  of  34764  dollars^ 
but  there  are  demands  ogainst  him  to  the  amount  of  14297 
dollars  ;  how  many  dollars  will  be  lelt  after  the  payment  of 
his  debts  ? 

24.  Four  men  bought  a  lot  of  land  for  482  dollars;  the 
first  man  paid  274  dollars,  the  second  man  194  dollars  lesf 
than  the  first,  and  the  third  man  20  dollars  less  than  the 
second  ;  how  much  did  the  second,  the  third,  and  the  fourth 
man  pay  ?  C  The  second  paid  80. 

A71S.  <  The  third  paid  60. 
f  The  fourth  paid  68. 

25.  A  man,  having  10,000  dollars,  gave  away  9  dollars ; 
how  many  had  he  left  ?  Aits.  9991. 


OF  SIMPLE  NUMBERS. 

IT  9.    1.  If  one  orange  costs  5  cents,  hov/  many  cents 

must  I  give  for   2  oranges  ?  how  many  cents  for  3 

orauges  ?  — —  lor  4  oranges  ? 

2.  One  bushel  of  apples  costs  20  cents ;  how  many  cent* 
must  I  give  for  2  bushels  ?  for  3  bushels  ? 


IT  9.  MULTIPLICATION    OF    SIMPLE    NUMBERS.  27 

3.  One  gallon  contains  4  quarts;  how  many  quarts  in  2 
gallons  ?  in  3  gullons  ?  in  4  gallons  > 

4.  Thnie  men  bought  a  horse ;  each  man  paid  23  doIVirs 
for  his  share ;  how  many  dollars  did  the  horse  cost  them  ? 

5.  A  man  has  4  farms  worth  324  dollars  each;  how  many 
dollars  are  they  all  worth  ? 

6.  In  one  dollar  there  are  one  hundred  cents ;  how  many 
cents  in  5  dollars  ? 

7.  How  much  will  4  pair  of  shoes  cost  at  2  dollars  a  pair? 

8.  How  much  will  two  pounds  of  tea  cost  at  43  cents  a 
pound  ? 

9.  There  are  24  hours  in  one  day  ;  how  many  hours  in  2 

days  ?     in  3    days  ?      in  4  days  ?     in  7 

days  ? 

10.  Six  boys  met  a  beggar,  and  gave  him  15  cents  each ; 
how  many  cents  did  the  beggar  receive  ? 

When  questions  occur,  (as  in  the  above  examples,)  where 
the  same  number  is  to  be  added  io  itself  se\eral  times,  the 
operation  may  be  mucli  lacililated  by  a  rule,  called  Multi' 
plication^  in  which  the  number  to  be  repeated  is  called  the 
multiplicand^  and  the  number  which  shows  how  many  times 
the  multiplicand  is  to  be  repeated  is  called  the  multiplier. 
The  multiplicand  and  multiplier,  when  spoken  of  collectively, 
are  called  the<^ac/or5,t  (producers,)  and  the  answer  is  called 
the  product, 

11.  There  is  an  orchard  in  which  there  are  5  rowsof  treps, 
and  27  trees  in  each  row;  how  many  trees  in  the  orchard  ? 

In   this  example,  it   is 

In  the  first  row,  27  trees,  evident    that    the   whole 

second  ....  27  nu'iiber  of  trees  will  be 

third     ....  27  equal   to  the  amount   of 

fourth  ....  27  fice  27's  added  together. 

fifth      ....  27  In    adding,    we     find 

_     _       -   ,         ,      , that    7  taken   five  times 

In  the  whole  orchard,  135  trees.  amounts  to  35.  We  write 

down  the  five  units,  and 
reserve  the  3  tens;  the  amount  of  2  taken  five  times  is  10, 
and  the  3,  which  we  reserved,  makes  13  which,  written  to 
the  left  of  units,  makes  the  whole  number  of  trees  135. 

If  we  have  learned  that  7  taken  5  times  amounts  to  35, 
and  that  2  taken  5  times  amounts  to  10,  it  is  plain  we  need 
write  the  number  27  but  once,  and  then,  setting  the  multi- 


t6  MULTIPLICATION   OF    SIMPLE    NUMBERS.       IT  9,  IQ. 

pli**r  under  it,  we  may  say,  5  times  7  aro  35,  writing  down 

the  5,  and  reserving  the  3   (tens)  as  in  add-tion.     Again,  5 

times  2    (tens)  are 

Multiplicand^')^  trees  in  each  row*  10,  (tens,)   and   3, 

Multipliers'^       brows,  (tens,)   v»hich    we 

n    J    .        ~7^  A  reserved,  make  13, 

ProdiK^t,    y  13o  trees,  Am.  (t^,,,^)  ^  before, 

T\  10.     12.  There  are  on  a  board  3  rows  of  spots,  and  4 
spots  in  each  row  ;  how  many  spots  on  the  board  ? 
%    %    ^    ^  A  slight  inspection   of  the  figure  will 

^  show,  that  the  number  of  fpots  may  be 
%  %  %  %  found  either  by  taking  4  thite  times,  (3 
#  *  *  #  times  4  are  12,)  or  by  taking  ^  Jour  times, 
(4  tunes  3  are  12 ;)  for  we  may  say  there 
are  3  rows  of  4  spots  each,  or  4  rows  of  3  spots  each*;  there- 
fore, we  may  use  either  of  the  given  numbers  for  a  multi- 
plier, as  best  suits  our  convenience.  We  generally  write 
the  numbers  as  in  subtraction,  the  larger  uppermost,  with 
units  under  units,  tens  under  tens,  &c.     Thus, 

Multiplicand,    4  spots.  Note,     4  and  3  are  the  factors^ 

Multiplier,        3  ro^.os.  which  produce   the   product  12. 

Product,  12  Ans, 

Hence,—^ Multiplication  is  a  short  way  of  performing  manff 
additions  ;  in  other  words,—//  is  the  method  of  repeating  any 
fl/uitttLiir  nny  given  number  of  times. 

SiGis.  ^wo  short  lines,  crossing  each  other  in  the  form 
of  the  letter  X,\are  the  sign  of  multiplication.  Thus,  3X4 
m  12,  signifies  that  3  times  4  are  equal  to  12,  or  4  times  3 
are  12. 

Note,  Before  any  progress  can  be  made  in  this  rule,  the 
following  table  must  be  committed  perfectly  to  memory. 


V  10.        MULTIPLICATION    OF    SIMPLE   NUMBERS. 


29    \ 


MULTIPLICATION   TABLE. 


2  times  0  are     0 

4  X  10  =  40 

7X    7  = 

49 

|10X    4=    40 

2X     1=    2 

4  X  11  =44 

7X    8  = 

56 

!10X    5=    50 

2  X    2—    4 

4  X  t2  =  48 

7X    9  = 

63 

10  X    6=    60 

2  X    3=    6 

5X    0=    0 
5X     1=    5 

5X    2  =  10 
5X    3  =  15 

7X  10  = 

70 

10  X    7=    70 

2X     4z=:    8 

7X11  = 

77 

10  X    8—    80 

2X     5rzzl0 

7X  12  = 

84 

10  X    9=    90 

2X     6z=12 

8  X    0  = 

0 

10  X  10  =  100 

2X    7=1=14 

5  X    4  =  20 

8  X     1  = 

8 

10  X  11  =  110 

2X     8zz:16 

5  X    5  =  25 

8X    2  = 

16 

10  X  12  =  120 

2  X    9  =  18 

5  X    6  =  30 

8X    3  = 

24 

11  X    0=      0 

2  X  10  zz:  20 
2  X  11=22 

5X     7  =  35 

5  X    8  =  40 

8X    4  = 
8X    5  = 

32 

40 

11  X  1=  11 
1 1  X    2  =    22 

2  X  12=z24 

5  X     9  =  45 

8X    6  = 

48 

11  X    3=    33 

3X     0:^    0 

5  X  10  =  50 

8X    7  = 

56 

1 1  X    4  =    44 

3X     1=    3 

5  X  11  =55 

8X    8  = 

64 

1 1  X    5  =    55 

3X    2=    0 

5  X  12  =  60 

8X    9  = 

72 

11  X    6=    66 

3X    3z=    9 

6X0—0 

8X10  = 

80 

11  X    7=    77 

3X    4=12 

6  X     1=6 

8X  11  = 

88 

11  X    8=    88 

3X    5=15 

6X    2—12 
3X    3  mis 

8X  12  = 

96 

11  X    9=    99 

3  X    6  =  18 

9X    0  = 

~0 

11  X  10=110 

3X    7  =  21 

6  X    4  zz=  24 

0X^1  = 

9 

11  X  11  =  121 

3  X    8  =  21 

OX     5  =  30 
QX    6zz:  36 

9X  *2  = 
9X    3  = 

18 
27 

11  X  12=132 

3X    9  =  27 

12  X  0=  0 
12  X  1=  12 
12  X  2=  24 
12  X    3=    36 

3  X  10  =  30 
3X  11=33 
3  X  12  =  36 

6  X    7  =  42 
6X    8  =  48 
6  X    9  =  54 

9X    4  = 
9X    5  = 
9X    6  = 

36 
45 
54 

4  X    0=    0 

6  X  10  =  60 

9X    7  = 

63 

12  X    4=    48 

4  X     1=4 

6  X  1 1  =  66 

9X    8  = 

72 

12  X    5=    60 

4  X    2=    8 

6  X  12  =  72 

9X    9  = 

81 

12  X    6  =    72 

4X    3=12 

7X    0=    0 

9X  10  = 

90 

12  X    7=    84 

4  X    4=10 

7X1=7 

9X  11  = 

99 

12  X    8=    96 

4X    5  =  20 

7X    2  =  14 
7X    3  =  21 

9X  12  = 

108 

12  X    9=108 

4X    6  =  24 

10  X    0  = 

0 

12  X  10=120 

4X    7z=:28 

7X    4  =  28 

10X1  = 

10 

12  X  11  =  132 

4X    8:x-32 

7X    5  =  35 

lOX    2  = 

20 

12  X  12=144 

4X    9  =  36 

7X    6  =  42 

10  X^  3  = 

30 

so  MULTIPLICATION   OF    SIMPLE    NUMBERS.         IT  10. 

9  X  2  zz:  how  many  ?  4  X  3  X  2  =  24. 

4  X  6  in  how  mail}'  ?  3x2X^1=:  how  many  ? 

8X  9=z  how  many  ?  7x1X2  =  how  many  ? 

3  X  7  iz:  how  many  ?  3x3x2  =  how  many  ? 

6  X  5  zz:  how  many  ?  3x2X4X5zz  how  many  ? 


13.  What  will  84  barrels  of  Hour  cost  at  7  dollars  a  bar- 
rel ?  Ans.  5S8  dollars. 

14.  A  merchant  bought  273  hats  at  8  dollars  each;  what 
did  they  cost?  Ans.  2184  dollars. 

15.  How  many  inches  are  there  in  253  feet,  every  foot 
being  12  inches  ? 

OiPERATlON.  The  product  of  12,  with  each  of  the  signifi- 

253  cant  ligures  or  digits,  having  been  commit- 

12  ted  to  memory  from  the  multiplication  table, 

A       on^  '^  ^'^  J"^^  ^^  *^^^^  ^^  multiply  by  1 J  as  by  a 

Am.  6[)6b  ^jj^gj^^  ^g^j^^^     rj-j^^g^  J  2  ^^^^^^  2  ^^^  3g^  ^^ 

16.  What  will  476  barrels  offish  cost  at  11  dollars  a  bar- 
rel ?  Am.  5236  dollars. 

^7.  A  piece  of  valuable  land,  containing  33  acres,  was 
sold  for  246  dollars  an  acre ;  what  did  the  whole  come  to  ? 
As  12  is  the  largest  number,  the  product  of  which,  with  the 
nine  digits,  is  found  in  the  multiplication  table,  therefore, 
when  the  multiplier  exceeds  r2,  we  multiply  by  each  figure 
in  the  multiplier  separately.     Thus  : 

OPERATION.  The  mullipli- 

2^6  dollars^  the  price  oj  I  acre.  ^j.  consists  of  3 

^  ''^'''^^''  ^/  «^'-^5-  tens  and  3  units. 

738  dollars,  the  price  of  3  acres.  ^^'^^^^^    raultiply- 

738    dollars,  the  price  of  30  acres.         ^"?     ^Y  .  ^^^     ^ 

. units     gives    us 

Ans.  8118  dollars,  the  price  of  33  acres.         738    dollars,  the 

price  of  3  acres. 
We  then  multiply  by  the  3  tens,  writing  the  first  figure  of 
tlie  product  (8)  in  ten'^s  place,  that  is,  directly  under  the  figure 
by  which  we  irndtiply.  It  now  appears,  that  the  product  by 
tlie  3  tens  consists  of  the  same  figures  as  the  product  by  the 
three  units  ;  but  there  is  this  difference — the  figures  in  the 
product  by  the  3  tens  are  all  removed  one  place  further  to- 
ward the  left  hand,  V  which  their  value  is  increased  ten- 
fold^ which  is  as  it  should  be,  because  the  price  of  30  acrci 


IT  10.        MULTIPLICATION    OF    SIMPLE    NUMBERS.  31 

is  e\ndently  ten  times  as  much  as  the  price  of  3  acres,  that 
is,  7880  dollars ;  and  it  is  plain,  that  these  two  products, 
added  together,  give  the  price  of  33  acres. 

These  examples  will  be  sufficient  to  establish  the  fol» 
lowing 

RULE. 

I.  Write  down  the  multiplicand,  under  which  write  the 
multiplier,  placing  units  under  units,  tens  under  tens,  &c., 
and  draw  a  line  underneath. 

II.  When  the  multiplier  does  not  exceed  12,  hegin  at  the 
right  hand  of  the  multiplicand,  and  multiply  each  figure  con- 
tained in  it  by  the  multiplier,  setting  down,  and  carrying,  as 
in  addition. 

III.  When  the  multiplier  exceeds  12,  multiply  hy  each 
figure  of  the  multiplier  j^eparately,  tirst  by  the  imitSy  then  by 
the  tens^  &c.,  remembering  always  to  place  the  first  figure  of 
each  product  directly  under  the  figure  by  which  you  multi- 
ply. Having  gone  through  in  this  manner  with  each  figure 
in  the  multiplier,  add  their  several  products  together,  and 
the  sum  of  them  will  be  the  product  required. 

EXAMPLES    FOR    PRACTICE. 

18.  There  are  320  rods  in  a  mile  ;  how  many  rods  are 
there  in  57  miles  ? 

19.  It  is  436  miles  from  Boston  to  the  city  of  Washing- 
ton ;  how  many  rods  is  it? 

20.  What  will  784  chests  of  tea  cost,  at  69  dollars  a 
chest  ? 

21.  If  1851  men  receive  758  dollars  apiece,  how  many 
dollars  will  they  all  receive  ?  Ans,   1403058  dollars. 

22.  There  are  24  hours  in  a  day;  if  a  ship  sail  7  miles  in 
an  hour,  how  many  miles  vvill  she  sail  in  1  day,  at  that  rate? 
how  many  miles  in  36  days  ?  how  many  miles  in  1  year,  or 
365  days  ?  Aiis.  6i320  miles  in  1  year. 

23.  A  merchant  hought  13  pieces  of  cloth,  each  piece 
containing  28  yards,  at  6  dollars  a  yard  ;  how  many  yards 
were  there,  and  what  was  the  whole  cost  ? 

Ans.  Tke  whole  cost  was  2184  dollars. 

24.  Multiply  37864  by  '235.  Product,       8.S98040. 

25 29831   ...      952 28399112. 

26 93956  ...    8704.  817793024. 


S3  CONTRACTIONS    IN   MULTIPLICATION.  IT  II. 


CONTRACTIONS  IN  MULTIPLICATION. 

I.  WTien  the  multiplier  is  a  composite  number* 
ir  11.  Any  number,  which  may  be  produced  by  the  mul- 
tiplication of  two  or  more  numbers,  is  called  a  composite 
number.  Thus,  15,  which  arises  from  the  multiplication  of 
5  and  3,  (5  X  3  rrr  15,)  is  a  composite  number,  and  the  num- 
bers 5  and  3,  which,  multiplied  together,  produce  it,  are  called 
component  parts ^  or  factors  of  that  number.  So,  also,  24  is  a 
composite  number ;  its  component  parts  or  factors  may  be  2 
and  12  (2  X  12  —  24 ;)  or  they  may  be  4  and  6  (4X6  = 
24  ;)   or  they  may  be  2,  3,  and' 4  (2  X  3  X  4  =^3  24.) 

1.  What  will  15  yards  of  cloth  cost,  at  4  dollars  a  yard? 

15  yards  are  equal  to  5  X  ^. yards.     The  cost  of  6 

4  yards  would  be  5  X  4  zz:  20  dollars  ;  and  because  15 

5  yards  contain  3  times  5  yards,  so  the  cost  of  15  yards 

—         will  evidently  be  3  times  the  cost  of  5  yards,  that  is, 

"^"         20  dollars  X  3  =:  60  dollars..  Ans,  60  dollars. 

o 

60 

Wherefore,  If  the  multiplier  be  a  composite  number^  we  may, 
if  we  please,  multiply  the  multiplicand  first  by  one  of  the  cowr- 
ponent  parts^  that  product  by  the  other ^  and  so  on^  if  the  com- 
ponent parts  be  more  than  two;,  and,  having  in  this  way 
multiplied  by  each  of  the  component  parts,  the  last  product 
will  be  the  product  required. 

2.  W^hat  will  136  tons  of  potashes  come  to,  at  96  dollan 
per  ton  ? 

8  X  12  1=  96.    It  follows,  therefore,  that  8  and   12  are 
component  parts  or  factors  of  96.     Hence, 
136  dollars,  the  price  of  1  ton. 

8  one  of  the  component  parts,  or  factors. 

1088  dollars,  the  price  of  8  tons. 

12  the  other  component  part,  or  factor. 


Ans,  13056  dollars,  the  price  of  96  tons. 

3.  Supposing  342  men  to  be  employed  in  a  certain  piece 
of  work,  for  which  they  are  to  receive  112  dollars  each, 
how  much  will  they  all  receive  ? 

8  X  7  X  2  r:=  112.  Ans.  38304  dollaw. 


IT  12,  13.     CONTRACTIONS    IN   MULTIPLICATION.  B$ 

4.  Multiply  3G7  by  48.  Product,  17616. 

6 853  ...  56.  47768. 

6 1086  ...  72.  78192. 

II.    Wien  the  multiplier  is  10,  100,  1000,  ^c. 

IT  12.  It  will  be  recollected,  (IT  3.)  that  any  figure,  on  be- 
ing removed  one  place  towards  the  left  hand,  has  its  value 
increased  tenfold ;  hence,  to  multiply  any  number  by  10,  it 
is  only  necessary  to  write  a  cipher  on  the  right  hand  of  it. 
Thus,  10  times  25  are  250 ;  for  the  5,  which  was  units  before, 
is  now  made  teits^  and  the  2,  v/hich  was  tens  before,  is  now 
made  hundreds.  So,  also,  if  any  figure  be  removed  two  places 
towards  the  left  hand,  its  value  is  increased  100  times,  &c. 
Hence, 

Wien  the  mtdtiplier  is  10,  100,  1000,  or  1  with  any  number 
of  ciphers  annexed^  annex  as  maiy  ciphers  to  the  multipli- 
cand as  there  are  ciphers  in  the  muitiplier,  and  the  m»ilti- 
plicand,  so  increased,  will  be  the  product  required.     Thus, 

Multiply  46  by      10,  the  product  is  460. 

*..83...     100,  83V.0. 

95  ...  1000,  95000. 

1  SAMPLES     FOR     PRACTICE. 

1.  What  wiil  70  barrels  of  flour  cost,  at  10  dollars  a  barrel  ? 

2.  If  100  men  receive  126  dollars  each,  how  many  dol- 
lars vvv!l  they  all  receive  ? 

3.  What  will  1000  pieces  of  broadcloth  cost,  estimating 
each  piece  at  312  d'^llars  ? 

4.  Multiply    5682  by     10000. 
6 82134  ...    100000. 

IT  13.  On  the  principle  suggested  in  the  last  IT,  it  follows, 
When  there  are  ciphers  on  the  right  hand  of  the  multipli- 
cand, multiplier,  either  or  both,  we  may,  at  first,  neglect 
these  ciphers,  multiplying  by  the  significant  fig^ircs  only; 
after  which  we  must  annex  as  many  ciphers  to  the  product 
as  there  are  ciphers  on  the  right  hand  of  the  multiplicand 
Mid  multiplier,  counted  together. 


S4  CONTRACTIONS    IN    MULTIPLICATION.  IT  13. 


EXAMPLES    FOR    PRACTICE. 

1.  If  1300  men  receive  460  dollars  apiece,  how  many 
dollars  will  they  all  receive  ? 

OPERATION.  '^^^  ciphers  in  the  multiplicand 

460  ^^^   multiplier,  cointed    together, 

1300  3.re  three.     Disregarding  these,  we 

.  write  the  sirjnificant  figures  ot"  the 

1^8  multiplier  under  the  significant  fig- 

46  ures  of  the  multiplicand,  and  multi- 

Ans.  198000  dollars.         ?/>;>  ^^^^/  which  we  annex  three 
ciphers  to   the  Mght  hand  of  the 
product, which  gives  the  true  answer. 

2.  The  number  of  distinct  buildings  in  New  England, 
appropriated  to  the  spinning,  weaving,  and  printing  of  cot- 
ton goods,  was  estimated,  in  1826,  at  400,  running,  on  an 
average,  700  sph.dles  each ;  what  was  the  whole  number  ol 
spindles  ? 

3.  Multiply    357  by  6300. 

4 8600  ....   17. 

5 9340  ....  460, 

6 5200  ....  410. 

7 378  ....  204. 

OPERATION. 
378 
204 
1522  I^^  the  operation  it  will  be  seen,  that  multi- 

000  l^b'i^g  by  ciphers  produces  nothing,     There- 

756  ^^^^' 

77112 

III.  Wheit  there  are  ciphers  between  the  significant  figures 
of  t4ie  multiplier y  we  may  omit  the  ciphers,  niultjplying  by 
the  significant  figures  onhj^  placing  the  first  figure  of  each  pro- 
duct directly  undei  the  tig'ire  by  which  we  multiply. 

EXAMPLES    FOR    PRACTICE. 

8.  Multiply  154326  by  3007. 


13»  SUPPLEMENT    TO    MULTIPLICATION.  35 

OrERATION. 

154326 
3007 


1080282 
462978 

Product,  464058282 


9.  Multiply     543  by      206. 

10 1620  ...     2103. 

11 36243  ...  32004. 


S^UPPZiSIMESNT 

TO  MULTIPLICATION. 

QUESTIONS. 

1.  What  is  multiplication  ?     2.  What  is  the  number  /o  be 

multiplied  called  ?     3.  to  multiply  by  called  ?     4.  What 

is  the  result  or  answer  called  ?  5.  Taken  collectively,  what 
are  the  multiplicand  ana  multiplier  called  ?  6.  What  is  the 
ngn  of  multiplication  ?  7.  What  does  it  show  ?  8.  In  what 
order  must  the  given  number  be  placed  for  multiplication  ? 

9.  How  do  you  proceed  when  the  multiplier  is  less  i\\diX\  12? 

10.  When  it  exceeds  12,  what  is  the  method  of  procedure  ? 

11.  What  is  a  composite  number?  12.  What  is  to  be  under- 
stood by  the  component  parts,  ov  factors,  of  any  number? 
13.  How  may  you  proceed  when  the  multiplier  is  a  compo- 
site number  1  14.  To  multiply  by  10,  100,  1000,  &c.,  what 
suffices?  15.  Why?  16.  When  there  are  ciphers  on  the 
right  hand  of  the  multiplicand,  multiplier,  either  or  both, 
how  may  we  proceed?  17.  When  there  are  ciphers  be- 
tween  the  significant  figures  of  the  multiplier,  hew  are  they 
to  be  treated  ? 

EXERCISES. 

1.  An  army  of  10700  men,  having  plundered  a  city,  took 
80  much  money,  that,  when  it  was  shared  among  them,  each 
man  received  46  dollars ;  what  was  the  sum  of  money 
taken? 


S6  SUPPLEMENT    TO    SIULTIPLICATIOTV.  IT   13. 

2.  Supposing  the  number  of  houses  in  a  ceitain  town  to 
be  145,  each  house,  on  an  average,  containing  two  families, 
ana  each  family  6  members,  what  would  be  the  number  of 
inhabitaixts  in  that  town  ?  Ans,  1740. 

3.  If  46  men  can  do  a  piece  of  work  in  60  days,  hovr 
many  men  will  it  take  to  do  it  in  one  day  ? 

4.  Two  men  depart  from  the  same  place,  and  travel  in 
opposite  directions,  one  at  the  rate  of  27  miles  a  day,  the 
other  31  miles  a  day;  hov/  far  apart  will  they  be  at  the  end 
of  6  days  ?  Ans.  348  miles. 

5.  What  number  is  that,  the  factors  of  which  are  4,  7,  6, 
and  20  ?  Ans,  3360. 

6.  If  18  men  can  do  a  piece  of  work  in  90  days,  how  long 
will  it  take  one  man  to  do  the  same  ? 

7.  Vv^hat  sura  of  money  must  be  divided  between  27 
men,  so  that  each  man  may  receive  115  dollars? 

S.  There  is  a  certain  number,  the  factors  of  which  are  89 
and  265  ;  what  is  that  number  ? 

9.  What  is  that  number,  of  which  9,  12,  and  14  are 
factors  ? 

10.  If  a  carriage  wheel  turn  round  346  times  in  running 
1  mile,  how  many  times  v/ill  it  turn  round  in  the  distance 
from  New  Yo.k  to  Philadelphia,  it  being  95  miles. 

Ans,  32870. 

11.  In  one  minute  are  60  seconds;  how  many  seconds  in 

4  minutes  ?   in  5  minutes  ?   in  20  minutes  ?   

in  40  minutes  ? 

12.  In  one  hour  are  60  minutes ;    how  many  seconds  in 

an  hour  ?    in  two  hours  ?      how  many  seconds  from 

nine  o'clock  in  the  morning  till  noon  ? 

13.  In  one  dollar  are  6  shillings;  how  many  shillings  in 

3  dollars  ?     in  300  dollars  ?     in  467  dollars  ? 

14.  Two  men,  A  and  B,  start  from  the  same  place  at  the 
same  time,  and  travel  the  same  y^'ay;  A  travels  52  miles  a 
day,  and  B  44  miles  a  day ;  how  far  apart  will  they  be  at 
the  end  of  10  days? 

15.  If  the  interest  of  100  cents,  for  one  yrar^  be  6  cents, 
how  many  cents  will  be  the  uiterest  for  2  years  ?    for 

4  years  ?  for  10  years  ?  for  35  years  ?  for  84 

years  ? 

16.  If  the  interest  of  one  dollar,  for  one  y^^ar,  be  six  cents, 

Wiat  is  the  interest  for  2  dollars  the  same  time  ?    5 

dollars  ?    7  dollars  ?    —  8  dollars  ?  95  doilar»l 


^  13,  14.        DIVISION   OF   SIMPLE   NTTMBERS.  87 

17.  A  farmer  sold  468  pounds  of  pork,  at  6  cents  a  pound, 
and  48  pounds  of  cheese,  at  7  cents  a  pound ;  kow  many 
cents  must  he  receive  in  pay  ? 

18.  A  boy  bought  10  oranges ;  he  kept  7  of  them,  and  sold 
^le  others  for  5  cents  apiece ;  how  many  cents  did  he  receive  ? 

19.  The  component  parts  of  a  certain  number  are  4,  5,  7, 
6,  9,  8,  and  3 ;    what  is  the  number  ? 

20.  In  1  hogshead  are  63  gallons;  how  many  gallons  in  8 
hogsheads  ?  In  1  gallon  are  4  quarts ;  how  many  quarts  in  8 
hogsheads  ?  In  1  quart  are  2  pints ;  how  many  pints  in  8  hogs- 
heads ? 


DIVISION 

OF  SIMPLE  NUMBERS. 

IT  14.  1,  James  divided  12  apples  among  4  boys  ;  how 
many  did  he  give  each  boy  ? 

2.  James  would  divide  12  apples  among  3  boys;  how 
many  must  he  give  each  boy  ? 

3.  John  had  15  ap})les,  and  gave  them  to  his  playmates,  who 
received  3  apples  each ;  how  many  boys  did  he  give  them  to  ? 

4.  If  you  had  20  cents,  how  many  cakes  could  you  buy 
at  4  cents  apiece  ? 

5.  How  many  yards  of  cloth  could  you  buy  for  30  dollars, 
at  5  dollars  a  yard  ? 

6.  If  you  pay  40  dollars  for  10  yards  of  cloth,  what  is  one 
yard  worth  ? 

7.  A  man  works  ^  ^J^  for  42  shillings;  how  many  shil- 
lings is  that  for  one  day  ? 

8.  How   many   quarts   in   4   pints?     in  6  pints? 

in  10  pints  ?  , 

9.  How  many  times  is  8  contained  in  88  ? 

10.  If  a  man  can  travel  4  miles  an  hour,  tow  many  houn 
would  it  take  him  to  travel  24  miles  ? 

11.  In  an  orchard  tliere  are  28  trees  standing  in  rows, 
9iid  there  ar^e  3  trees  in  a  row ;  how  many  rows  are  there  ? 

ReiTwrk.  When  any  on^s  thing  is  divided  into  two  equal 
parts,  one  of  thos^  parts  is  called  s^  half;  if  into  3  equal 
parts,  one  of  those  parts  is  called  a  third;  if  into  four  equal 
parts,  one  part  is  called  a  quarter  or  a  fourth;  if  into  liye, 
•ne  partis  called  a ///A,  and  so  on. 


38  DIVISION    OF    SIMRLE    NUMBERS.        M  14,  15* 

12.  A  boy  had  two  apples,  and  gave  one  half  an  apple  to 
each  of  his  companions  ;    how  many  were  his  companions  ? 

13.  A  boy  divided  four  apples  among  his  companions,  by 
giving  thorn  one  third  of  an  apple  each ;  among  how  many 
did  he  divide  his  apples? 

14.  Huw  many  quarters  m  3  oranges  ? 

15.  How  many  oranges  would  it  take  to  give  12  boys  one 
quarter  of  an  orange  each  ? 

16.  How  much  is  one  half  of  12  apples  ? 

17.  How  much  is  one  third  of  12  ? 
~n18.  How  much  is  one  fourth  of  12  ? 

li).  A  man  had  30  sheep,  and  sold  one  fifth  of  them ; 
how  many  of  them  did  he  sell  ? 

20.  A  man  purcliased  sheep  for  7  dollars  apiece,  and 
paid  for  ther.  ail  63  dollars;    what  was  their  numbei  ? 

21.  How  many  oranges,  at  3  cents  each,  may  be  bought 
for  12  cents  ? 

It  is  plain,  that  as  many  times  as  3  cents  can  be  taken 
from  12  cents,  so  many  oranges  may  be  bought;  the  object, 
therefore,  is  to  find  how  many  times  3  is  contained  in   12. 

12  cents. 

First  orangey     3  cents.  We  see  in  this  example,  thai 

— "  we    may   take   3    from    12   foui 

_,         ,  ^  times,  after  which  there  is  no  re 

Second  orangey  cents.         niainder;    consequently,  subtrac- 

Q  lion  alone  is  sufficient  for  the  ope- 

Tklrd  orange,  3  cents.         ^^^^0"  5    ^^^^  ^^'^  "^^y  ^^^^^  to  the 

—  same  result  by  a  process,  in  most 

3  cases  much   shorter,    called  Di- 

Fourth  orange y  3  cents.         vkion, 

0 

IT  15.  It  is  plain,  that  the  cost  of  one  orange,  (3  cents,) 
multiplied  by  the  number  of  oranges,  (4,)  is  equal  to  the 
cost  of  all  the  oranges,  (12  cents  ;)  12  is,  therefore,  a  pro- 
duct,  and  3  one  of  its  factors;  and  to  find  how  many  times 
3  is  contained  in  12,  is  to  find  the  other  factor,  which,  mul- 
tiplied into  3,  will  produce  12.  This  factor  we  find,  by 
trial,  to  be  4,  (4  X  3  zn  12;)  consequently,  3  is  contained  in 
12  4  times.  Ans.  4  oranges. 

22.  A  man  would  d\\  ide  12  oranges  equally  among  3  chil- 
dren;   how  many  oranges  Svould  each  child  have.? 

Here  the  object  is  to  divide  the  12  oranges  into  3  equal 


iri5. 


DIVISION   OF    SIMPLE    NUMBERS. 


89 


parts,  and  to  ascertain  the  number  of  oranges  in  each  of  those 
parts.  The  operation  is  evidently  as  in  the  last  example,  and 
consists  in  finding  a  number,  which,  multiplied  by  3,  will  pro- 
duce 12.     This  number  we  have  already  found  lo  be  4. 

Ans.  4  oranges  apiece. 

As,  therefore,  mullipllcation  is  a  short  way  of  performing 
many  additions  of  the  same  number ;  so,  dwisioii  is  a  short 
way  of  performing  many  subtractions  o^  the  same  number; 
and  may  be  defined.  The  method  of  finding  how  mmiy  limes 
one  number  is  cont aided  in  another ^  and  also  of  dividing  a  num- 
ber into  any  number  of  equal  parts.  In  all  cases,  the  process 
of  division  consists  in  finding  one  of  the  factors  of  a  given 
product,  when  tiie  other  factor  is  known. 

The  number  given  to  be  divided  is  called  the  dividend^ 
and  answers  to  the  product  in  multiplication.  The  number 
given  to  divide  by  is  called  the  divisor^  and  answers  to  ont  of 
the  factors  in  multiplication.  The  result^  or  answer  sought, 
is  called  the  quotient^  ffrnin  tlie  I^atin  word  quoties,  how 
many?)  and  answers  lo  the  other  factor. 

Sign.  The  sign  for  division  is  a  short  honzontal  line  be- 
tween two  dots,  -H.  It  shows  that  the  number  before  it  iu 
to  be  divided  by  the  number  after  it.  Thus  27  ~  9  =  3  is 
read,  27  divided  by  9  is  equal  to  3 ;  or,  to  shoridi  the  ex- 
pression, 27  by  9  is  3  ;  or,  9  in  27  3  times.  In  plpce  of  the 
dotSj  the  dividend  is  often  written  over  the  line,  and  the  di- 
visor under  it,  to  express  division ;  thus,  ^^  =  3,  read  as 
before. 


^^z=:3 
^z=9 


DM'ISIOJV 

TABLE 

.* 

f  =1* 

t=l 

J=l 

*=1 

S=i 

1=2* 

1=2 

1=2 

-y  =  2 

V  =  2 

f  =3 

f  =3 

J,^  =  3 

-^  =  3 

¥  =  3 

f  =4 

¥=--4 

JjS  =  4 

¥  =  4 

¥  =  4 

^  =  5 

^--=5 

^-  =  5 

¥  =  5 

s^~5 

-^  =  6 

V-  =  6 

^  =  6 

?Ji_6 

5/.  =  6 

4^  =  7 

¥  =  7 

^  =  7 

^  =  7 

¥  =  7 

4^  =  8 

¥  =  8 

\'=:8 

H'-  —  ^ 

*^  =  8 

■y:=9 

¥  =  9 

V  =  9 

V  =  9 

¥=9 

•  The  reading  used  hv  tho  jujpil  in  conimiiiin^  ihe  table  may  be^  2  by  2  U  I, 
4  by  2  is  2,  &c.  5  or,  2  in" 2  one  lime,  2  In  4  iwo  times,  &c. 


40 


DIVISION   OF   SIMPLE   NUMBERS.        IT  16,  16. 


DIVISION  TABLE— CONTINUED. 


t 
^ 

f 


.8.— 


1  =  1 

iS=l 

ii=l 

•^=2 

f*=2 

e=2 

^=3 

fS  =  3 

«-=3 

¥-=4- 

f5  =  4 

^  =  4 

V=5 

f8  =  5 

«=5 

^^=6 

fS  =  6 

«=6 

^=7 

t3  =  '7 

H=7 

^=8 

U  =  s 

ff=8 

V=9 

f8=9 

f*=9 

if  =  1 

ff  =2 

ff=3 
If  =4 
ff  =5 
e=6 
H=7 
If  =8 
Y/=9 


28  -h  7,  or  ^-  T=:  how  many  ? 
42  -f-  6,  or  ^(f-  zz:  how  many  ? 
54  -T-  9,  or  ^  =  how  many  ? 
82  -f-  8,  or  2^-  z=z  how  many  ? 
33  -7- 11,  or  ^^zz:  how  manj  ? 


49  -7-  7,  or  ^  z=  how  many  f 
32  -T-  4,  or  ^*i  zz:  how  many  ? 
99  -r- 1 1,  or  f  f  zz:  how  many  ? 
84  -:-  12,  or  ^^  =z  how  many? 
108 -f-  12,or\<^-zz:howmany? 


^  16«  23.  How  many  yards  of  cloth,  at  4  dollars  a  yard, 
can  be  bought  for  856  dollars  ? 

Here  the  number  to  be  divided  is  856,  which  therefore 
is  the  dimdend;  4  is  the  number  to  divide  by^  and  there- 
fore the  divisor.  It  is  not  evident  how  manj'  times  4  is  con- 
tained in  so  large  a  number  as  855.  This  difficulty  will  be 
readily  overcome,  if  we  decompose  this  number,  thus : 

856  :zz  800 +  40-1- 16. 
Beginning  with  the  hundreds,  we  readily  perceive  that  4  is 
contained  in  8  2  times;  consequently,  in  800  it  is  contained 
200  times.  Proceeding  to  the  tens,  4  is  contained  in  4  1 
time,  and  consequently  in  40  it  is  contained  10  times. 
Lastly,  in  16  it  is  contained  4  times.  We  now  have 
200 -j- 10 -[-4  ==214  for  the  quotient,  or  the  number  of 
times  4  is  contained  in  856.  Ans,  214  yards. 

We  may  arrive  to  the  same  result  without  decomposing 
the  dividend,  except  as  it  is  done  in  the  mind,  taking  it  by 
parts,  in  the  following  manner : 

For  the  sake  cf  convenience,  we 
write  down  the  dividend  with  the  divi- 
sor on  the  left,  and  draw  a  'ine  between 
them ;  we  also  draw  aline  underneath. 
Then,  beginning  on  the   left  hand,. 


Dividend, 
Divisor  J  4  )  856 

Quotient^      214 


IT  16.  mVlSlON    OF    SIMPLE    NUMBERS.  41 

we  seek  how  often  the  divisor  (4)  is  contained  in  8, 
(hundreds,)  the  left  hand  ligure;  finding  it  to  be  2  times, 
we  write  2  directly  under  the  8,  which,  falling  in  the  place 
of  hundreds,  is  in  reality  200.  Proceeding  to  tens,  4  is  con- 
tained in  5  (tens)  1  time,  which  we  set  down  in  ten^s 
place,  directly  under  the  5  (tens.)  But,  after  taking 4  times 
ten  out  of  the  5  tens,  there  is  1  ten  left.  This  1  ten  we  join 
to  the  6  units,  making  16.  Then,  4  into  16  goes  4  times, 
which  we  set  down,  and  the  work  is  done. 

This  mawner  of  performing  the  operation  is  called  Short 
Division,  The  computation,  it  may  be  perceived,  is  carried 
on  partly  in  the  mind,  which  it  is  always  easy  to  do  when 
the  divisor  does  not  exceed  12. 

RUL.E. 

From  the  illustration  of  this  example^  toe  derive  this  general 
rule  for  dividing^  when  the  divisor  does  not  exceed  12  : 

I.  Find  how  many  times  the  divisor  is  contained  in  the 
first  figure,  or  figures,  of  the  dividend,  and,  setting  it  direct- 
ly under  the  dividend,  carry  the  remainder,  if  any,  to  the 
next  figure  as  so  many  tens. 

II.  Find  how  many  times  the  divisor  is  contained  in  this 
dividend,  and  set  it  down  as  before,  continuiag  so  to  do  till 
all  the  figures  in  the  dividend  are  divided. 

Proof.  We  have  seen,  (IF  15,)  that  the  divisor  and  quo- 
tient are  factors,  whose  product  is  the  dividend,  and  we 
have  also  seen,  that  dividing  the  dividend  by  one  factor  is 
merely  a  process  for  finding  the  other. 

Hence  division  and  m-dilpUcation  mutually  prove  each  other. 

To  prove  division^  we  may  mulliphj  the  divisor  by  the  quo- 
tient, and,  if  the  work  be  right,  the  product  will  be  the  same 
as  the  dividend ;  or  we  may  divide  the  dividend  by  the  quo^ 
tientj  and,  if  the  work  is  right,  the  result  will  be  the  same  as 
tlie  divisor. 

To  prove  innllipUcation^  we  may  divide  the  product  by  one 
factor^  and,  if  the  work  be  right,  the  quotient  will  be  the  other 
factor,  ' 

EXAMPLES    FOR    PRACTICE. 

24.  A  man  would  divide  13,462,725  dollars  among  5  men  % 
how  many  dollars  would  each  receive  ? 
D* 


42  DIVISION    OF    SIMPLE    NUMBERS.        If  I&,  17, 

In  this  example,  as  we  cannot 
DMend        ^""^^  ^  '°  ^^^  '-^''^  '^Sure,  (1,)  we 
Divisor,  5 )  13,462,725      ^''i^f  *^°  Y'^''  ^"^  f,^-^'  ^  '°  ^^ 

'      '        '         '  \vi        crn    y?    timps.    and   fhprp    s\Tf^   !^ 


will   go  2  times,  and  there  are  3 
Quotienty       2,692,545      over,  which,  joined  to  4,   the  next 
figure,  makes  34  ;   and  5  in  34  will 
go  6  times,  &c. 
Proof.  In  proof  of  this  example,  we  mul- 

Quotient,  tiply  the  quotient  by  the  divisor, 

2,092,545  and,  as  the  product  is  the  same  as 

5  divisor.  the  dividend,  we  conclude  that  the 

1^469  72^  work  is  right.     From  a  bare  in- 

'       '  spection  of  the  above  example  and 

ifjs  proof,  it  is  plain,  as  before  stated,  that  division  is  the  re- 
verse of  multiplication,  and  that  the  two  rules  mutually  prove 
each  other. 

25.  How  many  yards  of  cloth  can  be  bought  for  4,354,560 

dollars,  at  2  dollars  a  yard  ?     at  3  dollars  ?     at 

4  dollars  ?     at  5  dollars  ?     at  6  dollars  ?     at 

7  ?     at  8  ?     at  9  ?     at  10  ? 

Note.     Let  the  pupil  be  required  to  prove  the  foregoing, 
and  all  following  examples. 

26.  Divide    1005903360  by  2,  3,  4,  5,  6,  7,  8,  9,  10,  11, 
and  12. 

27.  If  2  pints  make  a  quart,  how  many  quarts  in  8  pints  : 

in   12  pints  ?   in  20  pints  ?    in  24  pints  > 

in  248  pints  ? in  3764  pints  ? in  47632  pints  ? 

28.  Four  quarts  make  a  gallon ;  how  many  gallons  in  8 

quarts  ?   in  12  quarts  ?    in  20  quarts  ?   in  36 

quarts?     in  368  quarts  ?    in  4896  quarts  ?    

in  5436144  quarts? 

29.  A  man  gave  86  apples  to  5  boys ;  how  many  apples 
would  each  boy  receive  ? 

t)ividend.  Here,      dividing      the 

Divisor,   5  )  86  number    of    the    apples 

r\    *•    *       *TX     ,   ».       .    ,  (86)   by  the  number  of 

Quotient,  17-1  Remainder.  ^^ys,  (6,)  we  find,  that 
each  boy's  share  would  be  17  apples  ;  but  there  is  one  apple 
left. 

V17,     6)86  In  order  to  divide  all  the  apples  equal- 

— -        ly  among  the  boys,  it  is  plain,  we  must  di- 
* 'T        vide  this  one  remaining  apple  into  5  eqwd 


If  17.  DIVISION    OF    SIMPLE    JVUMBERS.  43 

parts^  and  give  one  of  these  parts  to  each  of  the  boys.  Then 
each  boy's  share  would  be  17  apples,  and  one  fifth  part  of 
another  apple;  which  is  written  thus,  17^  apples. 

A71S.  17 J  apples  eachc 
The  17,  expressing  whole  apples,  are  called  integers^  (that 
is,  whole  numbers.)  The  ^  (one  fifth)  of  an  apple,  express- 
ing part  of  a  broken  or  divided  apple,  is  called  a  fraction^ 
(that  is,  a  broken  number.) 

Fractions,  as  we  here  see,  are  written  v/ith  two  numbers, 
one  directly  over  the  other,  with  a  short  line  between  them, 
showing  that  the  vpper  number  is  to  be  divided  by  the 
lower.  The  upper  number,  ox  dividend^  is,  in  fractions,  call- 
ed the  numerator  J  and  the  lower  number,  or  divisor^  is  called 
the  denominator. 

Note,  A  number  like  17-J,  composed  of  integers  (17) 
iind  a  fraction,  (^,)  is  called  a  mixed  number. 

In  the  preceding  example,  the  one  apple,  which  was  left 
after  carrying  the  division  as  far  as  could  be  by  ivhole  num- 
bers, is  called  the  remainder^  and  is  evidently  a  part  of  the 
dividend  yet  undivided.  In  order  to  complete  the  division, 
this  remainder,  as  we  before  remarked,  must  be  divided  into 
5  equal  parts  ;  but  the  divisor  itself  expresses  the  number  of 
parts.  If,  now,  we  examine  the  fraction,  we  shall  see,  that 
it  consists  of  the  remainder  (1)  for  its  numerator^  and  the 
divisor  (5)  for  its  denominator. 

Therefore,  if  there  be  a  remainder^  set  it  down  at  the  right 
hand  of  the  quotient  for  the  numerator  of  a  fraction,  under 
which  write  the  divisor  for  its  denominator. 

Proof  of  the  last  example.  In  proving  this  example,  we 

17-i  find  it  necessary   to   multiply 

5  our  fraction  by  5 ;  but  this  is 

'~~  easily  done,  if  we  consider,  that' 

the  fraction  ^  expresses  one 
part  of  an  apple  divided  into  5  equal  parts ;  hence,  5  times 
^  is  1^=  1,  that  is,  one  whole  apple,  which  we  reserve  to  be 
added  to  the  unitSj  saying,  5  times  7  are  35,  and  one  we  re- 
served makes  36,  &c. 

30.  Eight  men  drew  a  prize  of  453  dollars  in  a  lottery  f 
\kovr  many  dollars  did  each  receive  } 


44  mViSIOJS    OF    SIMPLE    NUMBERS.         IT  IS,  19. 

Dividend,         Here,  after  carrying  the  division  as 
Divisor^  8  )  453  far  as  possible  by  whole  numbers,  we 

.  77"       have  a  remainder  of  5  dollars,  which, 

Quotient^  56f        written  d&  above  directed,  gives  for  the 

answer  56  dollars    and  |  (live  eighths)   of  another  dollar, 
to  each  man. 

TT  18.    Here  we  may  notice,  that  the  eighth  part  of  5  dol- 
lars is  the  same  as  5  times  the  eighth   part  of  1  dollar,  that 
is,  the  eighth  part  of  5  doilarh  is  §  of  a  dollar.     Hence,  f 
expresses  the  quotient  of  5  divided  by  8. 
Proof,  I  is  5  parts,  and  8  times  5  is  40,  that  is,  ^-  =:  5, 

66|-  which,  reserved  and  added  to  the  product  of  8  time* 

8  6,  makes  53,  &c.     Hence,  to  multiply  a  fractioUj 

-"—  we   may   multiply  the  mmeratOTy  and  divide  the 

^^  product  by  the  denominator. 

Or,  in  proving  division,  we  may  multiply  the  whole  num- 
ber in  the  quotient  onhj^  and  to  the  product  add  the  remain- 
der ;  and  this,  till  the  pupil  shall  be  more  particularly  taught 
in  fractions,  will  be  more  easy  in  practice.  Thus,  56  X  8 1= 
448,  and  448  +  5,  the  remainder,  z=  453,  as  before. 

31.  There  are  7  days  in  a  week ;  how  many  weeks  in 
365  days  ?  Ans,  52|  weeks. 

32.  When  flour  is  worth  6  dollars  a  barrel,  how  many 
barrels  mav  be  bought  for  25  dollars  /*  hov/  many  for  50  dol- 
lars ?  -'  for  487  dollars  ?  for  7631  dollars? 

33.  Divide  640  dollars  among  4  men. 

640  ^  4,  or  ^1^  =  160  dollars,  Ans. 

34.  G78  -^  6,  or  ^9-  rr=:  how  many  ?  Ans.  113. 

35.  ^^^  =z  how  many  ? 

36.  z.^^-^  =  ho\\  many? 

37.  34^ —  how  many?  Ans,  384f 

38.  2_zR^-=: how  many? 

39.  4(i.|jajLr=r:how  mauy  ? 

40.  afijijy?-X2rrr bow  many? 

IT  19-    41.  Divide  4370  dollars  equally  among  21  men. 

When,  as  in  this  example,  the  divisor  exceeds  12,  it  is 
evident  that  the  co.nputation  cannot  be  readily  carried  on  in 
the  mind,  as  in  the  foregoing  examples.  Wherefore,  it  is 
more  convenient  to  write  down  the  computation  at  lengthy 
in  the  following  manner : 


IT  19.  DIVISION   OF    SIMPLE    NUMBERS.  45 

OPERATION.  We  may  write  the  divisor 

Dlmsor.  DividemL  Quotient.         and  dividend  as  in  short  di- 

21  )  4370  (  20822^.  vision,  but,  instead  of  writing 

42  the  quotient  under  the  divi- 

-^ dend,  it  will  be  found  more 

^'^^  convenient  to  set  it  to  the 

^^  right  hand. 

'   2  Remainder.  taking  the   dividend  by 

parts,  we  seek  how  often  we 
can  have  21  in  43  (hundreds  ;)  rinding  it  to  be  2  times,  we 
set  do^vn  2  on  the  right  hand  of  the  dividend  for  the  high- 
est figure  in  the  quotient.  The  43  being  hundreds,  it  fol- 
lows, that  the  2  must  also  be  hundreds.  This,  however, 
we  need  not  regard,  for  it  is  to  be  followed  by  tens  and  units, 
obtained  from  ihe  tens  and  units  of  the  dividend,  and  will 
therefore,  at  the  end  of  the  operation,  be  in  the  place  of  hun- 
dreds, as  it  should  be. 

It  is  plain  that  2  (hundred) .  times  21  dollars  ought  now 
to  be  taken  out  of  the  dividend ;  therefore,  we  multiply  the 
divisor  (21)  by  the  quotient  figure  2  (hundred)  now  found, 
naking  42,  (hundred,)  which,  written  under  the  43  in  the 
dividend,  we  subtract,  and  to  the  remainder,  1,  (hundred,) 
bring  down  the  7,  (tens,)  making  17  tens. 

We  then  seek  how  often  the  divisor  is  contained  in  17, 
(tens ;)  finding  that  it  will  not  go,  we  write  a  cipher  in  the 
quotient,  and  bring  down  the  next  figure,  making  the  whole 
170.  We  then  seek  how  often  21  can  be  contained  in  170, 
and,  finding  it  to  be  8  times,  we  Avrite  8  in  the  quotient,  and, 
multiplying  the  divisor  by  this  number,  we  set  the  product, 
168,  under  the  170 ;  then,  subtracting,  Ave  find  the  remain- 
der to  be  2,  which,  written  as  a  fraction  on  the  right  hand 
of  the  quotient,  as  already  explained,  gives  208^^  dollars, 
for  the  answer. 

This  manner  of  performing  the  operation  is  called  Ij)ng 
Division,     It  consists  in  writing  down  the  whole  computation. 

From  the  above  example,  we  derive  the  following 

RULE. 

I.  Place  the  divisor  on  the  left  of  the  dividend,  separate 
them  by  a  line,  and  draw  another  line  on  the  right  of  the 
dividend  to  separate  it  from  the  quotient. 

II.  Take  as  many  figures,  on  the  left  of  the  dividend,  as 


46  DIYISIO  SIMt'LE    NUMBERS.  H   19. 

contain  the  divisor  once  or  more ;  seek  how  many  times  they 
contain  it,  and  place  the  answer  on  the  right  hand  of  the 
di^adend  for  the  first  figure  in  the  quotient. 

III.  Multiply  the  divisor  by  this  quotient  figure,  and  write 
the  product  under  that  part  of  the  dividend  taken. 

IV.  Subtract  the  product  iVom  the  figures  above,  and  to  the 
remainder  bring  down  the  next  figure  in  the  divitlend,  and 
divide  the  number  it  makes  up,  as  before.  So  continue  to 
do,  till  all  the  figures  in  the  dividend  shall  have  been  brought 
down  and  divided. 

Note  1.  Having  brought  down  a  figure  to  the  remainder, 
if  the  number  it  makes  up  be  less  than  the  divisor,  write 
a  cipher  in  the  quotient,  and  bring  down  the  next  figure. 

Note  2.  If  the  product  of  the  divisor,  by  any  quotient 
figure,  be  greater  than  the  part  of  the  dividend  taken,  it  is  an 
evidence  that  the  quotient  figure  is  too  large^  and  must  be 
diminished.  If  the  remainder  at  any  time  be  greatn  than 
the  divisor,  or  equal  to  it,  the  quotient  figure  is  too  snially  and 
must  be  increased. 

EXAMPLES   FOR  PRACTICE. 

1.  How  many  hogsheads  of  molasses,  at  27  dollars  a  hogs- 
head, may  be  bought  for  6318  dollars  ? 

Ans,  234  hogsheads, 

2.  If  a  man's  income  be  1248  dollars  a  year,  how  much 
is  that  per  week,  there  being  52  weeks  in  a  year  ? 

Ars.  24  dollars  per  week. 

3.  What  will  be  the  quotient  of  153598,  divided  by  29  ? 

Ans.  5296^1- 

4.  How  many  times  is  63  contained  in  30131  ? 

Ans,  478^  J  times  ;  that  is,  478  tiroes,  and  JJ  of  another 
time. 

5.  What  v/ill  be  the  several  quotients  of  7652,  divided  by 
16,  23,  34,  86,  and  92? 

6.  If  a  farm,  containing  256  acres,  be  worth  7168  dollars, 
what  is  that  per  acre  ? 

7.  Wh.*t  will  be  the  quotient  of  974932,  divided  by  365  ? 

Ans.  2671  ^Vs- 

8.  Divide  3228242  dollars  equally  among  563  men  ;  how 
many  dollars  must  each  man  receive  ?         Ans.  5734  dollars. 

9.  If  57624  be  divided  into  216,  586,  and  976  equal  parts, 
what  will  be  the  magnitude  of  one  of  each  of  these  equal 
parts'* 


IT  20,  21.  CONTRACTIONS    IN    DIVISION.  47 

Ans.  The  magnitude  of  one  of  the  kst  of  these  equal  parts 
will  be  59^\% 

10.  How  many  times  does  1030603615  contain  3215? 

Alls.  320561  times. 

11.  The  earth,  in  its  annual  revolution  round  the  sun,  is 
said  to  travel  596088000  miles  ;  what  is  that  per  hour,  there 
being  8766  hours  in  a  year  ? 

12.  JL2.3i.i6  2.8aii  ::^  how  mauy  ? 

13.  Axixoa^k  r=  how  many? 

14.  5^6_<^9^^L  _  1^0 w  many  ? 


CONTRACTIONS  IN  DIVISION. 
I.   When  the  divisor  is  a  composite  number. 

IT  20.  1.  Bought  15  yards  of  cloth  for  60  dollars ;  how 
much  wa>  that  per  yard  ? 

15  yai^s  are  3  X  5  yards.  If  there  had  been  but  5  yajds, 
the  cost  of  one  yard  would  be  -^^-  -:=  12  dollars  ;  but,  as  there 
are  3  times  5  yards,  the  cost  of  one  yard  will  evidently  be 
but  one  third  part  of  12  dollars ;  that  is,  ^  =  4  dollars.  Ans. 

Hence,  when  the  divisor  is  a  composite  number,  we  may, 
if  we  please,  divide  the  dividend  by  one  of  the  component 
parts,  and  the  quotient ^  arising  from  that  division,  by  the 
other  :  the  last  quotient  will  be  the  answer. 

2.  If  a  man  can  travel  24  miles  a  day,  how  many  days 
will  it  take  him  to  travel  264  miles  ? 

It  will  evidently  take  him  as  many  days  as  264  contains  24. 
OPERATION. 
24  =  6X4.  6)264  24)264(11  days,  iln«. 

—  24 

11  days.  24 

3.  Divide  576  by  48=  (8  X  6.) 

4.  Divide  1260  by  63=  (7  X  9.) 

5.  Divide  2430  by  81. 
C.  Divide  448  by  56. 

II.   To  divide  by  10,  100,  1000,  &c. 
IT  ai.    1.  A  prize  of  2478  dollars  Is  owned  by  10  men, 
is  each  man's  share  ? 


48  CONTRACTIONS    IN   DIVISION.  Tr21,22. 

Each  man's  share  will  be  equal  to  the  number  of  tens  con- 
tained in  the  whole  sum,  and,  if  one  of  the  figures  be  cut  off 
at  the  right  hand,  all  the  figures  to  the  left  may  be  consid- 
ered so  manv  tens;  therefore,  each  man's  share  will  be 
247f^  dollars." 

It  is  evident,  also,  that  if  2  figures  had  been  cut  off  from 
the  right,  all  the  remaining  figures  would  have  been  so  ma- 
ny hundreds;  if  3  figures,  so  many  thousands,  &c.  Hence 
we  derive  tbis  general  Rule /or  dividing  by  10,  100,  1000, 
&c.  :  Cut  off  from  the  right  of  the  dividend  so  many  figures 
as  there  are  ciphers  in  the  divisor ;  the  figures  to  the  left 
of  the  point  Xvill  express  the  quotient,  and  those  to  the  right, 
the  remainder. 

2.  In  one  dollar  are  100  cents ;  how  many  dollars  in  42400 
cents?  'Ans,  424  dollars. 
494'00          Here  the  divisor  is  100;   we  therefore  cut  off  2 

'  figures  on  the  right  hand,  and  all  the  figures  to  the 

left  (424)  express  the  dollars. 

3.  How  many  dollars  in  34567  cents  ? 

Ans.  345^^  dollars. 

4.  How  many  dollars  in  4567840  cents? 

5.  How  many  dollars  in  345600  cents  ? 

6.  How  many  dollars  in  42604  cents  ?  Ans.  426^^^. 

7.  1000  mills  make  one  dollar  ;  how  many  dollars  in  4000 
mills  ?     in  25000  mills  ?     in  845000  ? 

8.  How  many  dollars  in  b487  mills?     Ans.  6-/^^^  dollars. 

9.  Hov/  many  dollars  in  42863  mills  ?     in  368456 

mills  ?     in  96842378  mills  ? 

10.  In  one  cent  are  10  mills;    how  many  cents  in  40 

mills  ?   in  400  mills  ?     in  20  mills  ?     in  468 

mills  ?     in  4784  mills  ?     in  34640  mills  ? 

III.    When  there  are  ciphers  wi  the  right  hand  of  the  divisor. 

^  CtSi.    1.  Divide  480  dollars  among  40  men  ? 

In  this  example,  our  divisor, 

^InNdftlrf^'^^^^'  (^^?)   ^^   ^   composite   number^ 

4|u;^4S|U  (10  X  4  =r  40 ;)  we  may,  there- 

12  dolls.  Ans.  ^or^j  divide  by  one   component 

part,  (10,)  and  that  quotient  by 

the  other,  (4;)  but  to  divide  by  10,  we  have  seen,  is  but  ta 

cut  off  the  right  hand  figure,  leaving  the  figures  to  the  left 


^ 


IT  52.  SUPPLEMENT  TO  BITISION.  49 

of  the  point  for  the  quotient,  which  we  diyide  hy  4,  and  the 
work  is  done.  It  is  evident,  that,  if  our  diviwsor  had  been 
400,  we  should  have  cut  off  2  figures,  and  have  divided  in 
the  same  manner ;  if  4000, 3  figures,  &;c.  Hence  this  gene- 
ral Rule  : —  When  there  are  ciphers  at  the  right  hand  of  the  di- 
tlsor^  cut  them  off,  and  also  as  many  places  in  the  dividend ; 
divide  the  remaining  figures  in  the  dividend  by  the  remain- 
ing figures  in  the  divisor;  then  annex  tlie  figures,  cut  off 
from  the  dividend,  to  the  remainder. 

2.     Divide  748346  by  8000. 
Dividend* 

Dirwor,  8|000)748|346. 

Quotient^  93. — 1346  Remainder.         An$.  93|3«S8. 
?.  Divide  46720367  by  4200000. 
Dividend. 
42[00000)467|20367(ll3^^  Quoiienf, 

"47  /      ''""'' 

42  f  UNW' 

520367  Rmainder. 

4.  How  many  yards  of  cloth  can  be  bought  for  340500 
dollars,  at  20  dollars  per  yard  ? 

5.  Divide  76428400  by  900000. 

6.  Divide  345006000  by  84000. 

7.  Divide  4680000  by  20,  200,  2000,  20000,  300,  4000, 
50,  600,  70000,  and  80. 


QUESTIONS. 

1.  What  is  division  ?  2.  In  what  does  tlyb  prccfim  ctf  dl» 
vision  consist  ?  3.  Division  is  the  r€«>ef5^  of  what  ?  4.  >Vhdt 
k  the  number  to  he  divided  called,  and  to  what  does  it  aw- 
Fwer  in  multiplication  ?  5.  What  is  the  number  to  divide 
by  called,  and  to  what  does  it  answer,  &c.  ?  6.  What  is  the 
result  or  answer  called,  &c.  ?  7.  What  is  the  siffn  of  divi 
•ion,  and  what  does  it  show  ?  8.  What  is  the  other  way  of 
expressing  division  ?  9.  What  is  shiyrt  dimim^  and  how  it 
E 


50  SUPPLEMENT    TO   DIVISION.  If  2% 

it  performed?  10.  How  is  division  proved!  11.  Howls 
Tnultiplication  proved?  12.  What  are  integers^  or  v/bole 
numbers?  13.  What  are  fractions,  or  broken  numbers? 
14.  What  is  a  mixed  number  ?  15.  When  there  is  any  thing 
left  after  division,  what  is  it  called,  and  ho\/  is  it  to  be 
written?     16.  How    are    fractions   written!     17.  What   is 

the  upper  number  called?      18.  the  lower  number? 

19.  How  do  you  multiply  a  fraction  ?  20.  To  what  do  the 
numerator  and  the  denominator  of  a  fraction  answer  in  di- 
vision ?  21.  What  is /ow^r  division  ?  22.  Rule?  23.  When 
the  divisor  is  a  composite  number,  how  may  we  proceed  ? 
24.  When  the  divisor  is  10,  100,  1000,  &:c.,  how  may  the 
operation  be  contracted  ?  25.  When  there  are  ciphers  at 
the  right  hand  of  the  divisor,  how  may  we  proceed  ? 

EXERCISES. 

1..  An  army  of  1500  men,  having  plundered  a  city,  took 
2625000  dollars  ;  what  was  each  man's  share? 

2.  A  certain  number  of  men  were  concerned  in  the  pay*- 
ment  of  18950  dollars,  and  each  man  paid  25  dollars ;  what 
was  the  number  of  men  ? 

3.  If  7412  eggs  be  packed  in  34  baskets,  how  many  in  a 
basket  ? 

4.  What  number  must  I  midtiply  by  135  that  the  pro- 
duct may  be  505710? 

5.  Light  moves  with  such  amazing  rapidity,  as  to  pass 
from  the  sun  to  the  earth  in  about  the  space  of  8  minutes^ 
Admitting  the  distance,  as  usually  computed,  to  be  95,000,000 
miles,  at  what  rate  per  minute  does  it  travel  ? 

6.  If  the  product  of  two  numbers  be  704,  and  the  multi- 
plier be  11,  what  is  the  multiplicand  ?  Aiis,  64. 

7.  If  the  product  be  704,  and  the  multiplicand  64,  what 
is  the  multiplier  ?  Am.  11. 

8.  The  divisor  is  18,  and  the  dividend  144;  what  is  the 
quotient  ? 

9.  The  quotient  of  two  numbers  is  8,  and  the  dividend 
144  ;  what  is  the  divisor  ? 

10.  A  man  wishes  to  travel  585  miles  in  13  days ;  how 
far  must  he  travel  each  day? 

11.  If  a  man  travels  45  miles  a  day,  in  how  many  day* 
tnll  he  travel  585  miles  ? 


fr  22.  SUPPLEMENT    TO    DIVISION.  51 

12.  A  man  sold  35  cows  for  560  dollars ;  how  much  was 
that  for  each  cow  ? 

13.  A  man,  selling  his  cows  for  IC  dollars  each,  received 
for  all  560  dollars ;  how  many  did  he  sell  ? 

14.  If  12  inches  make  a  foot,  how  many  feet  are  there  in 
364812  inches? 

15.  If  364812  inches  are  30401  feet,  how  many  inches 
make  one  foot  ? 

16.  If  you  would  divide  48750  dollars  among  50  men, 
how  many  dollars  would  you  give  to  each  one  ? 

17.  If  you  distribute  48750  dollars  among  a  number  of 
men,  in  such  a  manner  as  to  give  to  each  one  975  dollars, 
how  many  men  receive  a  share  ? 

18.  A  man  has  17484  pounds  of  tea  in  186  chests;  how 
many  pounds  in  each  che?t  ? 

19.  A  man  would  put  up  17484  pounds  of  tea  into  chests 
containing  94  pounds  each ;  how  many  chests  must  he  have? 

20.  In  a  certain  town  there  are  1740  inhabitants,  and  12 

persons  in  each  house  ;  how  many  houses  are  there  ? in 

each  house  are  2  families ;  how  many  persons  in  each  family? 

21.  If  2760  men  can  dig  a  certain  canal  in  one  day,  how 
many  days  would  it  take  46  men  to  do  the  same  ?  How 
many   men   would  it  take  to  do  the    work  in   15    days  ? 

in  5  days  ?     ■ in  20   days  ?     in  40  days  ? 

in  120  days? 

22.  If  a  carriage  wheel  turns  round  32870  times  in  run- 
ning from  New  York  to  Philadelphia,  a  distance  of  95  miles, 
how  many  times  does  it  turn  in  running  1  mile  ?    Ans.  346. 

23.  Sixty  seconds  make  one  minute  ;  how  many  minutes 

m  3600  seconds  ?     in  86400  secoiids  r     — —  in  604800 

seconds  ?     in  2419200  seconds  ? 

24.  Sixty  minutes  make  one  hour;    how  many  hours  in 

1440  minutes  ?     in   10080  minutes  ?     in  40320 

minutes  ? in  525960  minutes  ? 

25.  Twenty-four  hours  make  a  day;  how  many  days  in 
168  hours  ?     ' in  672  hours  ?     in  8766  hovrs  ? 

26.  How  many  times  can  I  subtract  forty-eight  from  four 
hundred  and  eighty  ? 

27.  How  many  timea  3478  is  equal  to  47854  ? 

28.  A  bushel  of  grain  is  32  quarts ;  how  many  quarts  must 
I  dip  out  of  a  chest  of  grain  to  make  one  half  (^)    of  a 

bushel  ?   for  one  fourth  (J)  of  a  bushel  ?  for  one 

eighth  (})  of  a  bushel  ?  Aits,  to  the  last^  4  quarts. 


50  MISCEttANEOUS    QUESTIONS*  IT  22,  23v 

29.  How  many  is  i  of  20?     J  of  48  ? j-of 

247  ?     ^  of  345678  ?     ^  of  204030648  ? 

^7i5.  «o  </j^  last^  102015324- 

30.  How  many  walnuts  are  one  third  part  (^)  of  3  wal- 
nuts ?    ^  of  6  walnuts  ? ^  of  12  ?     i  of  30  > 

^  of  45  ?     i  of  300  ?     ^  of  478  ?     f 

of  3456320  ?  Ans.  to  the  last,  1 152106^. 

31.  Whatisiof4?   1  of 20  ?   i  of  320?   J 

of  7843  ?  Ans,  to  the  last,  1960|. 


MXS0£3Z.Z.ANE0nS   QUESTIONS, 

Involving  the  Principles  of  the  preceding  Rules. 

Note.  The  preceding  rules,  viz.  Numeration,  Addition, 
Subtraction,  Multiplication,  and  Division,  are  called  the  Fun- 
damental Rules  of  Arithmetic,  because  th^y  are  the  foun- 
dation of  all  other  rules. 

1.  A  man  bought  a  chaise  for  218  dollars,  and  a  horse  for 
142  dollars;  what  did  they  both  cost  ? 

2.  If  a  horse  and  chaise  cost  360  dollars,  and  the  chaise 
cost  218  dollars,  what  is  the  cost  of  the  horse  ?  If  the  horse 
cost  142  dollars,  what  is  the  cost  of  the  chaise? 

3.  If  the  sum  of  2  numbers  be  487,  and  the  greater  num- 
ber be  348,  what  is  the  less  number?  If  the  less  number 
be  139,  what  is  the  greater  number? 

4.  If  the  minuend  be  7842,  and  the  subtrahend  3481, 
what  is  the  remainder  ?  If  the  remainder  be  4361,  and  the 
minuend  be  7842,  what  is  the  subtrahend? 

^  23*  When  the  minuend  and  the  subtrahend  are  given, 
how  do  you  find  the  remainder? 

When* the  minuend  and  remainder  are  given,  how  do  you 
find  the  subtrahend  ? 

When  the  subtrahend  and  the  remainder  are  given,  how 
do  you  find  the  minuend  ? 

When  you  have  the  sum  of  two  numbers,  and  one  of  them 
given,  hov/  do  you  find  the  other  ? 

When  you  have  the  greater  of  two  numbers,  and  their 
difference  given,  how  do  you  find  the  less  number  ? 

When  you  have  the  less  of  two  numbers,  and  their  differ- 
eace  given,  how  do  you  and  the  greater  number  ? 


If  23,  24.  MISCEr^LANEOUS    QUESTIONS.  53 

6.  The  sum  of  two  numbers  is  48,  and  one  of  the  numbers 
is  19 ;  what  is  the  other  1 

6.  The  greater  of  two  numbers  is  29,  and  their  difference 
10  ;    what  is  the  less  number? 

7.  The  less  of  two  numbers  is  19,  and  their  difference  is 
10 ;  what  is  the  greater  ? 

8.  A  man  bought  5  pieces  of  cloth,  at  44  dollars  a  piece; 
974  pairs  of  shoes,  at  3  dollars  a  pair ;  600  pieces  of  calico, 
at  6  dollars  a  piece  ;  what  is  the  amount  ? 

9.  A  man  sold  six  cows,  worth  fifteen  dollars  each,  and  a 
yoke  of  oxen,  for  67  dollars ;  in  pay,  he  received  a  chaise, 
worth  124  dollars,  and  the  rest  in  money;  how  much  money 
did  he  receive  ? 

10.  What  will  be  the  cost  of  15  pounds  of  butter,  at  13 
cents  per  pound  ? 

11.  How  many  bushels  of  wheat  can  you  buy  for  487 
dollars,  at  2  dollars  per  bushel  ? 

IT  24.  When  the  price  of  one  pound,  one  bushel,  &c.  of 
any  commodity  is  given,  how  do  you  find  the  cost  of  any 
number  of  pounds,  or  bushels,  &c.  of  that  commodity  ?  If 
the  price  of  the  1  pound,  &c.  be  in  cents,  in  what  will  the 

whole  cost  be?     If  in  dollars,  what?     if  in  shillings? 

if  in  pence  ?    &c. 

When  the  cost  of  any  given  number  of  pounds,  or  bushels, 
&c.  is  given,  how  do  you  find  the  price  of  one  pound  or 
bushel,  &c.     In  what  kind  of  money  will  the  answer  be  ? 

When  the  cost  of  a  number  of  pounds,  &e.  is  given,  and 
also  the  price  of  one  pound,  &c.,  how  do  you  ijttd  the  num- 
ber of  pounds,  &c. 

12.  When  rye  is  84  cents  per  bushel,  what  will  be  the  cost 
of  948  bushels  ?  how  many  dollars  will  it  be  ? 

13.  If  648  pounds  of  tea  cost  284  dollars,  (that  is,  28400 
cents,)  what  is  the  price  of  one  pound  ? 

When  the  factors  are  given,  how  do  you  find  the  product? 

When  the  product  and  one  factor  are  given,  how  do  you 
find  the  other  factor  ? 

When  the  divisor  and  quotient  are  given,  how  do  you 
find  the  dividend  ? 

When  the  dividend  and  quotient  are  given,  how  do  yoti 
find  the  divisor  ? 

14.  What  is  the  product  of  754  and  25  ? 

E* 


5t 


MISCELLANEOUS    QUESTIONS.  1(  ^4,  25. 


15.  What  numoer,  multiplied  by  25,  will  produce  18850? 

16.  What  number,  multiplied  by  754,  will  produce  18650  ? 

17.  If  a  man  save  six  cents  a  day,  how  many  cents  would 

he  save  in  a  year,   (365   days,)  ?     hov/  many  in  45 

years  ?    how  many  dollars  would  it  be  ?    how  many  cows 
could  he  buy  with  the  money,  at  12  dollars  each  ? 

Ans,  to  the  last^  82  cows,  and  1  dollar  50  cents  remainder. 

18.  A  boy  bought  a  number  of  apples;  he  gave  away  ten 
of  them  to  his  companions,  and  afterwards  bought  thirty-four 
more,  and  divided  one  half  of  what  he  then  had  among  four 
companions,  who  received  8  apples  each  ;  how  many  apples 
did  the  boy  first  buy  > 

Let  the  pupil  take  tlie  last  number  of  apples,  S,  and  re- 
verse the  process.  Ans,  40  apples. 

19.  There  is  a  certain  number,  to  which,  if  4  be  added, 
and  7  be  subtracted,  and  the  difference  be  multiplied  by  8, 
and  the  product  divided  by  3,  the  quotient  will  be  64  ;  what 
is  that  number  ?  Ans.  27. 

20.  A  clicss  board  has  8  rows  of  8  squares  each ;  how 
many  squares  on  the  board  ? 

TT  S5.  21.  There  is  a  spot  of  ground  5  rods  long,  and  3 
rods  wide  ;  how  many  square  rods  does  it  contain  ? 

Note,  A  square  rod  is  a 
square  (like  one  of  those  in 
the  annexed  figure)  meas- 
uring a  rod  on  each  side. 
By  an  inspection  of  the 
figure,  it  will  be  seen,  that 
there  are  as  many  squares 
in  a  row  as  rods  on  one  side, 
and  that  the  number  of  rows 

is  equal  to  the  number  of  rods  on  the  other  side ;   therefore, 

5  X  3=  15,  the  number  of  squares. 

Ans,  15  square  rods. 

A  figure  like  A,  B,  C,  D,  having  its  opposite  sides  equal 
and  parallel,  is  called  a  parallelogram  or  oblong, 

22.  There  is  an  oblong  field,  40  rods  long,  and  24  rods 
wide ;  how  many  square  rods  does  it  contain  ? 

23.  How  many  square  inches  in  a  board  12  inches  long, 
and  12  inches  broad  ?  Ans,  144. 


r> c 

!  I 

f I 

-"         ■  i  I 

._L_^^ 1 

_  J.  J 

A  B 


t   25.  MISCELLANEOUS    Q^UESTIONS.  55h 

24.  How  many  square  feet  in  a  board  14  teet  long  and  2 
feet  wide  ? 

25.  A  certain  township  is  six  miles  square ;  how  many 
square  miles  does  it  contain  ?  Ans,  36. 

26.  A  man  bought  a  farm  for  22464  dollars ;  he  sold  one  half 
of  it  for  12480  dollars,  at  the  rate  of  20  dollars  per  acre;  how 
many  acres  did  he  buy  ?  and  what  did  it  cost  him  per  acre  ? 

27.  A  boy  bought  a  sled  for  86  cents,  and  sold  it  again  for 
8  quarts  of  walnuts ;  he  sold  one  half  of  the  nuts  at  12  cents 
a  quart,  and  gave  the  rest  for  a  penknife,  which  he  sold  for 
34  cenis ;  how  many  cents  did  he  lose  by  his  bargains  ? 

28.  In  a  certain  school-house,  there  are  5  rows  of  desks ; 
m  each  row  are  six  seats,  and  each  seat  will  accommodate 
2  pupils ;  there  are  also  2  rows,  of  3  seats  each,  of  the 
same  size  as  the  others,  and  one  long  seat  where  S  pupils 
may  sit;  how  many  scholars  will  this  house  accommo- 
date? '  Alts.  80. 

29.  How  many  square  feet  of  boards  will  it  take  for  the 
floor  of  a  room  16  feet  long,  and  15  feet  wide,  if  we  allow 
12  square  feet  for  waste  ? 

30.  There  is  a  room  6  yards  long  and  5  yards  wide ;  how 
many  yards  of  carpeting,  a  yard  wide,  will  be  sufficient  to  co\  er 
the  floors,  if  the  hearth  and  fireplace  occupy  3  square  yards  ? 

31.  A  board,  14  feet  long,  contains  28  square  feet;  what 
IS  its  breadth  ? 

32.  How  many  pounds  of  pork,  worth  6  cents  a  pound, 
can  be  bought  for  144  cents  ? 

33.  How  many  pounds  of  butter,  at  15  cents  per  pound, 
must  be  paid  for  25  pounds  of  tea,  at  42  cents  per  pound  ? 

34.  4  +  5+6+1+8=  how  many ? 

35.  4  +  3+10  —  2  —  4+6  —  7  =  how  many ? 

36.  A  man  divides  30  bushels  of  potatoes  among  3  poor 
men  ;  how  many  bushels  does  each  man  receive  ?  What  is 
J  of  thirty  ?     How  many  are  §  {two  thirds)  of  30  ? 

37.  How  many  are  one  third    (^)    of  3  ?     • of  6  ? 

of  9  ?     of  282  ?     of  45674312  ? 

38.  How  many   are  tivo  thirds    (f )    of  3  ?     of  6  ? 

of  9?     of  282?     of  45674312? 

39.  How  many  are  J  of  40  ?     f  of  40  ?     ■ ^  of 

60?   fof60?    lofSO?     of  124?     of 

246876  ?     f  of  246876  ? 

40.  How  many  is -i  of  80?     |of80?   ^  of  100? 

41.  An  inch  is  one  twelfth  part  i-^^)  of  a  foot;  how  many 


56  COMPOUND    NUMBERS.  IT  25,  26 

feet  in  12  inches  ?     in  24  inches  ?     in  36  inches  ? 

in  12243648  inches  ? 

42.  If  4  pounds  of  tea  cost  128  cents,  what  does  1  pound 

cost  ?     2  pounds  ?     3  pounds  ?     5  pounds  ? 

100  pounds  ? 

43.  When  oranges  are  worth  4  cents  apiece,  how  many 
can  be  bought  for  four  pistareens,  (or  20  cent  pieces  ?) 

44.  The  earth,  in  moving  round  the  sun,  travels  at  the 
rate  of  68000  miles  an  hour  ;  how  many  miles  does  it  travel 
in  one  day,  (24  hours  ?)  how  many  miles  in  one  year,  (365 
days  ?)  and  how  many  days  would  it  take  a  man  to  travel 
this  last  distance,  at  the  rate  of  40  miles  a  day  ?  how  many 
years  ?  Aiis.  to  the  last,  40800  years. 

45.  How  much  can  a  man  earn  in  20  weeks,  at  80  cents 
per  day,  Sundays  excepted  ? 

46.  A  man  married  at  the  age  of  23 ;  he  lived  with  his 
wife  14  years ;  she  then  died,  leaving  him  a  daughter,  12 
years  of  age ;  8  years  after,  the  daughter  was  married  to  a 
man  5  years  older  than  herself,  who  was  40  years  of  age 
when  the  father  died ;  how  old  was  the  father  at  his  death  ? 

Ano.  60  years 

47.  There  is  a  field  20  rods  longj  and  8  rods  mide  ;  how 
many  square  rods  does  it  contain  t  Ans,  160  rods. 

48.  What  is  the  width  of  a  field,  which  is  20  rods  long, 
and  contains  160  square  rods  ? 

49.  What  is  the  length  of  a  field,  8  rods  wide,  and  con- 
taining 160  square  rods  ? 

50.  What  is  the  width  of  a  piece  of  land,  25  rods  long, 
and  containing  400  square  rods  ? 


IT  26.  VA  number  expressing  things  of  the  same  kind  is 
called  a  simple  number ;'  thus,  100  men,  56  years,  75  cents, 
are  each  of  them  simple  numbers  Y  but  when  a  number  ex- 
presses things  of  difterent  kinds,'4t  is  called  a  compound  nuwr 
her ;  thus,  43  dollajs  25  cents  and  3  mills,  is  a  compound 
number;  so  4  years  6  months  and  3  days,  46  pounds  7 
shillings  and  6  pence,  are  compound  numbers. 

NoteS  Different  kinds,  or  names,  are  usually  called  <f«/- 
fereni  denominations. 


V  £6.  FEDERAL    MONEY.  57 


FEDERAL  MONEY. 

"^Federal  money  is  the  coin  of  the  United  States.  The 
xinds,  or  denominations,  are  eagles,  dollars,  dimes,  cents, 
and  mills." 

10  mills      -        -         -         are  equal  to  -  1  cent 

10  cents,  (=100  mills,)  -         -         -         =1  dime. 

10  dimes,  (  =  100  cents  =  1000  mills,)  -         zzi  1  dollar. 

10dollars,(=100dimes  =  1000cents  =  10000miris)rz:l  eagle.* 

Sign.  '  This  character,  $ ,  placed  before  a  number,  shows 
it  to  express  federal  money. 

As  10  mills  make  a  cent,  10  cents  a  dime,  10  dimes  a 
dollar,  &c.  it  is  plain,\that  the  relative  value  of  mills,  cents, 
dimes,  dollars  and  eap;les  corresponds  to  the  orders  of  units, 
tens,  hundreds,  &c.  in  simple  numbers.  Hence,  they  may 
be  read  either  in  the  lowest  denomination,  or  partly  in  a 
higher^  and  partly  in  the  lowest  denomination*     Thus  : 

•^  2  o5  ^    . 

34  6  52  may  be  read,  34652  mills ;  or  3465  cents  and  2  mills ; 
or,  reckoning  the  eagles  tens  of  dollars,  and  the  dimes  tern 
of  cents,  which  is  the  usual  practice,  the  whole  may  be 
read,  34  dollars  65  cents  and  2  mills. 

For  ease  in  calculating,  a  point  (')  called  a  separatrix^'\ 
is  placed  between  the  dollars  and  cents,  showing  that  all  the 
figures  at  the  left  hand  express  dollars,  while  the  two  first 
figures  at  the  right  hand  express  cents,  and  the  third^  mills. 
Thus,  the  above  example  is  written  $  34'652  ;  that  is,  34 
dollars  65  cents  2  mills,  as  above.  \  As  100  cents  make  a 
dollar,  the  cents  may  be  any  number  from  1  to  99,  often  re- 
quiring two  figures  to  express  them  ;  for  this  reason,  two 
places  are  appropriated  to  cents,  at  the  right  hand  of  the 
point,  and  if  the  number  of  cents  be  less  than  ten^  requiring 
but  one  figure  to  express  them,  the  ten'^s  place  must  be  filled 
with  a  cipher.  Thus,  2  dollars  and  6  cents  are  written  2*06. 
10  mills  make  a  cent,  and  consequently  the  mills  never  ex- 
ceed 9,  and  are  always  expressed  by  a  single  figure.     Only 

•  The  eagle  is  a.^old  coin,  the  dollar  anrl  dime  arc  silver  coins,  ihe  cent  is  a 
copper  coin.  The  mill  is  only  imao^imry ,  there  heing"  no  coin  of  that  denomina- 
tion.    There  are  half  eagles,' half  dollars,  half  dimes,  and  half  c-ents,  real  coins: 

t  The  character  used  for  the  sqnratnx,  in  the  "  Scholars'  Arithmetic/'  was 
the  comma}  the  comma  inverted  is  here  adopted,  to  distinguish  it  from  the  conv- 
ma  used  in  punctuation. 


58         REDUCTION  OF  FEDERAL  MONEY.   U  26,  27. 

(me  place,  therefore,  is  appropriated  to  mills,  that  is,  the 
place  irnmedidtely  following  cents,  or  the  third  place  from 
the  point.  When  there  are  no  cents  to  be  written,  it  is  evi- 
dent that  we  must  write  tioo  ciphers  to  fill  up  the  places  of 
cents.  Thus,  2  dollars  and  7  mills  are  written  2^007.  Six 
cents  are  written  '06,  and  seven  mills  are  written  '007. 

Note.  Sometimes  5  mills  ^=  ^  a  cent  is  expressed  frac- 
tionally: thus,  425  (twelve  cents  and  five  mills)  is  ex- 
pressed 12|,  (twelve  and  a  half  cents.) 

17  dollars   and   8  mills    are   written,   17'008 

4  dollars  and  5  cents,     -----     4^05 
75  cents,     ---------       '75 

24  dollars, 24' 

9  cents, '09 

4  mills, '004 

6  dollars  1  cent  and  a  mills,    -    -    -    C'013 

Write  down  470  dollars  2  cents ;  342  dollars  40  cents 
and  2  mills ;  100  dollars,  1  cent  and  4  mills ;  1  mill ;  2 
mills;  3  mills;  4  mills;  ^  cent,  or  5  mills ;  1  cent  and  1  mill  5 
2  cents  and  3  mills ;  six  cents  and  one  mill ;  sixty  cents  and 
one  mill ;  four  dollars  and  one  cenl ;  three  cents ;  five  cents; 
nine  cents. 


REDUCTION  OF  FEDERAL  MONEY. 

IT  27.    How  many  mills  in  one  cent  ?     in  2  cents  ? 

in  3  cents  ?     in  4  cents  ?     in  6  cents  ?    in  9 

cents  ?     in  10  ceDts  ?     in  30  cents  ?     in  78 

cents  ?   in  100  cents,  (=  1  dollar)  ?  in  2  dollars  ? 

in  3  dollars?     in  4  dollars  ?    in  484  cents  ? 

in  563  cents  ?     —  in  1  cent  and  2  mills  r     in  4 

cents  and  5  mills  ? 

How  many  cents  in  2  dollars  ?   in  4  dollars  ?   in 

8  dollars  ?  in  3  dollars  and  15  cents  ?  in  5  dol- 
lars and  20  cents  ?     in  4  dollars  and  6  cents  ? 

How  many  dollars  in  400  cen^s  ?     in  600  cents  ? 

in  380  cents  ?     in  40765  cents  ?     How  many 

cents  in   1000  mills  ?     How  many  dollars  in  1000  mills  ? 

in  3000  mills  ?     in  8000  mills  ?     in  4378 

mills  ?    in  846732  mills  ? 

This  changing  one  kind  of  money ^  §"0.  into  another  kindy  with* 
oiU  altering  the  value,  is  called  Reduction. 


IT  2T,  28.        ADDITION    OF   FEDERAL   MONEY.  59 

As  there  are  10  mills  in  one  cent,  it  is  plain  that  cents  are 
changed  or  reduced  to  mills  by  multiplying  them  by  10,  tha 
is,  by  merely  annexing  a  cipher,  (TT  12.)  100  cents  make  a 
dollar ;  therefore  dollars  are  changed  to  cents  by  annexing  2 
ciphers,  and  to  mills  by  annexing  3  ciphers.  Thus,  16  dollars 
=  1600  cents  izz  16000  mills.  Again,  to  change  mills  back 
to  dollars,  we  have  only  to  cut  off  the  three  right  hand 
figures,  (IT  21  ;)  and  to  change  cents  to  dollars,  cut  off  the 
two  right  hand  figures,  when  all  the  figures  to  the  left  w411  be 
dollars,  and  the  figures  to  the  right,  cents  and  mills. 

Reduce  34  dollars  to  cents.  Ans,  3400  cents, 

Reduce  240  dollars  and  14  cents  to  cents. 

Ans.  24014  cents. 

Reduce  $  748443  to  mills.  Ans,  74S143  mills. 

.    Reduce  748143  mills  to  dollars.  Ans,  $  748443. 

Reduce  3467489  mills  to  dollars.  Ans.  3467^489. 

Reduce  48742  cents  to  dollars.  Ans.  $  487*42. 

Reduce  1234678  mills  to  dollars. 

Reduce  3469876  cents  to  dollars. 

Reduce  $  4867'467  to  mills. 

Reduce  984  mills  to  dollars.  Ans.  $  '984 

Reduce  7  mills  to  dollars.  Ans.  $  '007 

Reduce  $  '014  to  mills. 

Reduce  17846  cents  to  dollars. 

Reduce  984321  cents  to  mills. 

Reduce  961 7J  cents  to  dollars.  Ans.  $93=*^i7^. 

Reduce  2064^  cents,  503  cents,  106  cents,  921J  cents, 
500  cents,  726]  cents,  to  dollars. 

Reduce  86753  mills,  96000  mills,  6042  mills,  to  dollars. 


ADDITION   AND    SUBTRACTION   OF   FEDERAL 
MONEY. 

^  28.  JBVom  what  has  been  said,  it  is  plain,  that  we  may 
readily  reduce  any  sums  in  federal  money  to  the  same  de- 
nomination, as  to  cents,  or  mills,  and  add  or  subtract  them 
ajs  simple  numbers.  Or,  what  is  the  same  thing,  we  may 
tet  down  the  sums,  taking  care  to  write  dollars  under  dollarSy 
cents  tinder  cents,  and  mills  under  mills,  i^  such  order,  that  ths 
teparating  points  of  the  several  numbers  shall  fall  directly  under 
9ach  other,  and  add  them  up  as  simple  numbers,  placing  th^ 
t^aaratrix  in  the  amount  directly  under  the  other  points* 


40                   SUBTRACTIOJf    OF    FEDERAL    MONEY.  U   28, 

What  is  the  amount  of    $487^643,   $  132'007,  $4<04, 

md  $  264402  ?  Ans.  $  887*792. 

OPERATION.  OPERATION. 

487643  mills.  or,                  $487^643 

132007  mftls.     '  $132*007 

4040  mills.  $      4*04 

264102  mills.  $264*102 


Mount,  887792  mills,  =:  $  887*792.  $  887*792  Amount, 

EXAMPLES    FOR    PRACTICE. 

1.  Bought  1  barrel  of  flour  for  6  dollars  15  cents,  10 
pounds  of  coffee  for  2  dollars  30  cents,  7  pounds  of  sugar 
for  92  cents,  1  pound  of  raisins  for  12|  cents,  and  2  oranges 
for  6  cents;    what  was  the  whole  amount.'^     Ans.  $  10*155. 

2.  A  man  is  indebted  to  A,  $  237*62  ;  to  B,  $  350  ;  to  C, 
$86*12^;  to  D,  $9*62^;  and  to  E,  $0*834;  what  is  the 
amount  of  his  debts  ?  Ans,    $  684*204. 

3.  A  man  has  three  notes  specifying  the  following  sums, 
viz.  three  hundred  dollars,  fifty  dollars  sixty  cents,  and 
nine  dollars  eight  cents ;  what  is  the  amount  of  the  three 
notes?  Ans,  $359*68. 

4.  What  is  the  amount  of  $56*18,  $7*37^,  $280, 
$0*287,    $17,  and  $90*413.? 

5.  Bought  a  pair  of  oxen  for  $76*50,  a  horse  for  $85, 
and.a  cow  for  $  17*25  ;  what  was  the  whole  amount.? 

6.  Bought  a  gallon  of  molasses  for  28  cents,  a  quarter  of 
tea  for  37^-  cents,  a  pound  of  salt  petre  for  24  cents,  2  ya^ds 
of  broadcloth  for  11  dollars,  7  yards  of  flannel  for  1  dollar 
62^  cents,  a  skein  of  silk  for  6  cents,  and  a  stick  of  twist  for 
4  cents ;  how  much  for  the  whole  ? 


SUBTRACTION  OF  FEDERAL  MONEY. 

7.  A  man  gave  4  dollars  75  cents  for  a  pair  of  boots,  and 
2  dollars  12J  cents  for  a  pair  of  shoes ;  how  much  did  th^ 
boots  cost  him  more  than  the  shoes  ? 

OPERATION.  OPERATION. 

4750  mills.  or,  $  4*75 

2125  mills.  $2*125 

2625  mills  =  $  2*625  Ans.        $  2*625  Ans, 


ir  29.        MULTIPLICATiaX    GF   FEDERAL   MONEY.  61 

8.  A  itfan  bought  a  cow  for  eighteen  dollars,  and  sold  her 
again  for  twenty-one  dollars  thirty-seven  and  a  half  cents  ; 
how  much  did  he  gain  r  Am,  $  3 '375. 

9.  A  man  bought  a  horse  for  82  dollars,  and  sold  him 
again  for  seventy-nine  dollars  seventy-five  cents  ;  did  he  gain 
or  lose  ?  and  how  much  ?  Ans,  He  lost  $  2^25. 

10.  A  merchant  bought  a  piece  of  cloth  for  $  1*76,  which 
proving  to  have  been  damaged,  he  is  willing  to  lose  on  it 
$  16^50 ;  what  must  he  have  for  it  ?  Ans.  $  159^50. 

11.  A  man  sold  a  farm  for  $5400,  which  was  $  725^37^ 
more  than  he  gave  for  it ;  what  did  he  give  for  the  farm  ? 

12.  A  man,  having  $  500,  lost  S3  cents  j  how  much  had 
he  left? 

13.  A  man's  income  is  $  1200  a  year,  and  he  spends 
$  800*35  ;  how  much  does  he  lay  up  ? 

14.  Subtract  half  a  cent  from  seven  dollars. 

15.  How  much  must  you  add  to  $  16*82  to  r^ake  $25  ? 

16.  How  much  must  you  subtract  from  $250,  to  leave 
$  87*14  ? 

17.  A  man  bought  a  barrel  of  flour  for  $6*25,  7  pounds 
of  coffee  for  $1*41;  he  paid  a  ten  dollar  bill;  how  much 
must  he  receive  back  in  change  ? 


MULTIPLICATION  OF  FEDERAL  MONEY. 

II  29.    1.  What  will  3  yards  of  cloth  cost,  at  $4*62 J-  a 
yard  ? 
OPERATION.  $  4*625  are  4625  mills,  which 

$   4*625  multiplied  by  3,  the  product  is 

3  13875  mills.     13875  mills  may 

^"TTT^T^     t  ^^^  ^^   reduced   to   dollars  by 

$  13*875,  the  answer,  placing  a  point  between  the  third 
and  fourth  figures,  that  is,  between  the  hundreds  and  thou- 
sands, which  is  pointing  off  as  many  places  for  cents  and 
mills,  in  the  product,  as  there  are  places  of  cents  and  mills 
in  the  sum  given  to  be  multiplied.  This  is  evident ;  for,  as 
1000  mills  make  1  dollar,  consequently  the  thousandi  in 
13875  mills  must  be  so  many  dollars.     - 

2.  At  16  cents  a  pound,  what  will  123  pounds  of  butter 
cost? 

F 


62  MULTIPLICATION    OF   FEDERAL    MONEY.        TT  29. 

OPERATION.  -^^  ^^^  product  of 

123,  the  number  of  pounds.  any   two    numbers 

16  cents ^  the  price  per  pound,  '^^'^^\  ^^  ^^   same, 

whichever  of  them 

"738  be  made  the  multi- 

123  plier,  therefore  the 

^imsllhemmver.  quantity,  being  the 

'  larger    number,    is 

made  the  multiplicand,  and  the  price  the  multiplier. 

123  times  16  cents  is  1963  cents,  which,  reduced  to  dollars, 
is  $19^68. 

RULE. 

From  the  foregoing  examples  it  appears,  that  thii  multi- 
plication of  federal  money  does  not  differ  from  the  multipli- 
cation of  simple  numbers.  The  jiroduct  will  he  the  answer  in 
the  lowest  denomination  contained  in  the  given  sum^  w^hich  may 
then  be  reduced  to  dollars. 

EXAMPLES    FOR    PRACTICE. 

3.  What  will  250  bushels  of  rye  come  to,  at  $  0^88^  per 
bushel?  Ans.  $221 '25. 

4.  What  is  the  value  of  87  barrels  of  flour,  at  $  6'37^  a 
barrel  ? 

5.  What  will  be  the  cost  of  a  hogshead  of  molasses,  con- 
taining 63  gallons,  at  28^  cents  a  gallon  ?       Ans.  $  17'955. 

6.  If  a  man  spend  12^  cents  a  day,  w^hat  wnll  that  amount 
to  in  a  year  of  365  days  ?  what  will  it  amount  to  in  5 
years  ?     *  Ans.  It  will  amount  to  $228'12J  in  5  years, 

7.  If  it  cost  $36^75  to  clothe  a  soldier  1  year,  how  much 
will  it  cost  to  clothe  an  army  of  17800  men  ? 

Ans.  $654150. 

8.  Multiply  $  367  by  46. 

9.  Multiply  $  0^273  by  8600. 

10.  What  will  be  the  cost  of  4848  yards  of  calico,  at  25 
cents,  or  one  quarter  of  a  dollar,  per  yard?         Ans.  $1212. 

Note.  As  25  cents  is  just  J  of  a  dollar,  the  operation  in 
the  above  example  may  be  contracted,  or  made  shorter ;  for, 
at  one  dollar  per  yard,  the  cost  would  be  as  many  dollars  as 
there  are  yards,  that  is,  $  4848 ;  and  at  one  quarter  (^)  of  a 
dollar  per  yard,  it  is  plain,  the  cost  would  be  one  quarter  (J) 
ts  many  dollars  as  there  are  yards,  that  is,  ^-^=  $1212. 


^29.         MULTIPLICATION    OF    FEDERAL    MONEY.  63 

When  one  quantity  is  contained  in  another  exactly  2, 3, 4, 
6,  &c.  times,  it  is  called  an  aliquot  or  even  part  of  that  quantir 
ty ;  thus,  25  cents  is  an  aliquot  part  of  a  dollar ,  because  4  times 
25  cents  is  just  equal  to  1  dollar ;  and  6  pence  is  an  aliquot 
part  of  a  shilling,  because  2  times  six  pence  just  make  1 
shilling.  The  following  table  exhibits  some  of  the  aliquot 
parts  of  a  dollar : 

TABLE.  From  the  illustration  of  the  last 

ctsV  example,  it  appears,  that,  when  the 

50    •=!  ^  of  a  dollar,         price  per  yard^  pound^  &c.  is  one  of 

33^  =  i  o/  a  dollar,         these  aliquot  parts  of  a  dollar,  the 

25    zzi^ofa  dollar,         cost  may  be  found|  by  dividing  the 

20    z=:  ^  of  a  dollar,         given  number  of  yards,  pounds,  &c. 

12^=1  lofa  dollar,         ^Y  ^^^^  number  which  it  takes  of 

^.        1     r     J  n  the  price  to  make  1  dollar.  \  If  the 

Qi- =1  ^ir  of  a  dollar,  •  ^  i.     ,-^        x  j*  -V   i 

IT  1     r     J  77  price  be  50  cents,  we  divide  by  2  ; 

5    =^oJ  a  dollar,         ^^  ^6  cts.  by  4 ;  if  12^  cts.  by  8,  &e. 

This  manner  of  calculating  the  cost  of  articles,  by  taking 

aliquot  parts,  is  usually  called  Practice, 

11.  What  is  the  value  of  14756  yards  of  cotton,  cloth,  at 
12^  cents,  or  ^  of  a  dollar,  per  yard  ? 

By  practice.  By  multiplication, 

8)14756  14756 

425 


Am.   $1844^50 


73780 
29512 
14756 


$  1844'500  Ans,  as  before. 

12.  What  is  the  cost  of  18745  pounds  of  tea,  at  $  ^50,  zr:^ 
dollar,  per  pound  ?  ■  Ans.   $  9372*50 

13.  What  is  the  value  of  9366  bushels  of  potatoes,  at  SS^J 
cents,  or  -^  of  a  dollar,  per  bushel  ?        SJ^^  zn  $  3122  Ans. 

14.  What  is  the  value   of  48240  pounds  of  cheese,  at 
$  *06J,  zz:  -Jg-  of  a  dollar,  per  pound  ?  A71S,  $  3015. 

15.  What  cost  4870  oranges,  at  5  cents,  =  5^  of  a  dollar, 
apiece?  Ans,  $243*50 

16.  What  is  the  value  of  151020  bushels  of  apples,  ai  20 
cents,  =:  ^  of  a  dollar,  per  bushel  ?  Ans,  $  30204. 

17.  What  will  264  pounds  of  butter  cost,  at  12^  cents 
per  pound?  Ans.  $33. 

18.  What  cost  3740  yards  of  cloth,  at  $  1*25  per  yard  ? 


64  MULTIPLICATION    OF    FEDERAL    l^IONEY.         IT  30a 

4)  $  3740  =z  cost  at  $  1^  per  yard. 
935  1=  cost  at  $    '25  per  yard. 

Ans.  $4675==  cost  at  $  1'25  per  yard. 
19.  What  is  the  cost  of  8460  hats,  at    $1*12^  apiece? 

at  $  1'50  apiece  ?      at  $  3^20  apiece  ?      at 

$  4^06^  apiece  ? 

A71S,    $9517^50.      $12690.      $27072.      $  34368<75.  ^ 

TT  30.     To  find  the  value  of  articles  sold  by  the  100,  or  1000. 

1.  y/hat  is  the  value  of  865  feet  of  timber,  at  $5  per 

hundred  ? 

Were    the   price    $5 
OPERATION.  p„  j^^f^   ^^   j/p,^j„^  fjjg 

^  value    would   be    865  X 

I  $5z=z  $  4325  ;    but  the 

$  4325  =1  value  at  $5  per  foot,     price  is  $  5  for  100  feet ; 

consequently,  $  4325  is 
100  times  the  true  value  of  the  timber  ;  and  therefore,  if  we 
divide  this  number  ( $  4325)  by  100,  we  shall  obtain  the 
true  value ;  but  to  divide  by  100  is  but  to  cut  oil'  the  two 
right  hand  figures,  or,  in  federal  money,  to  remove  the  scpara- 
trix  two  figures  to  the  left.  Aiis.  $  43'25. 

It  is  evident,  that,  were  the  price  so  much  per  thousand^ 
the  same  remarks  would  apply,  with  the  exception  of  cutting 
off  three  figures  instead  of  two.  Hence  we  derive  the 
general  Rule  for  finding  the  value  of  articles  sold  by  the  100, 
or  1000  :v-Multiply  the  number  by  the  price,  and,  if  it  be 
reckoned  by  the  100,  cut  off  the  two  right  hand  figures,  and 
tlie  product  will  be  the  answer,  in  the  same  kind  or  denomi-  . 
nation  as  the  price.  If  the  article  be  reckoned  by  the  1000, 
cut  off  the  three  right  hand  figures., 

exabipl.es  for  practice. 

2.  What  is  the  value  of  4250  feet  of  boards,  at  $  14  per 
1000  ?  Ans.  69  dollars  and  50  cents. 

OPERATION, 
4250 

$  14  In  this  example,  because  the  price  is  at 

"TZTTT  so  much  per  1000  feet^  Vv'e  divide  by  1000,. 

^250  ^^  ^^^  off  three  figures. 

$  59*500 


IF  80,  31.         DIVISION   OF    FEDERAL    MONEY.  65 

3.  What  will  3460  feet  of  timber  come  to,  at    $  4  per 
hundred  ? 

4.  What  will  24650  bricks  come  to,  at  5  dollars  per  1000  ? 
6.  What  will  4750  feet  of  boards  come  to,  at  $  12^25  per 

1000  ?  Am.  58487. 

6.  What  will  38600  bricks  cost,  at  $4^75  per  1000  ? 

7.  What  will  46590  feet  of  boards  cost,  at  $  10'625  per 
^'lOOO? 

8.  What  will  75  feet  of  timber  cost,  at  $  4  per  100  ? 

9.  What  is  the  value  of  4000  bricks,  at  3  dollars  per  1000  ? 


DIVISION  OF  FEDERAL  MONEY. 
IT  31.  1.  If  3  yards  of  cloth  cost  $5^25,  what  is  that  a  yard? 

OPERATION  f^'^^   }^    ^^^    ^^^*^> 

3\g(25  which  divided  by  3,  the 

^ quotient    is     175    cents, 

Answer^  175  cents^  =  $  V75,         which,  reduced  to  dollars, 

is  $  1'75,  the  answer. 

2.  Bought  4  bushels  of  corn  for   $  3 ;  what  was  that  a 
bushel  ? 

4  is  not  contained  in  3 ;  we  may,  however,  reduce  tU^ 
(S3  to  cents,  by  annexing  two  ciphers,  thus : 
OPERATION.  300  cents  divided  by  4,  the  quotient 

4)3^  |g  Y5  cents,  the  price  of  each  bush,  of 

Am,      <75  cents,        corn. 

3.  Bought  18  gallons  of  brandy,  for  $42^75  ;  what  did  it 
cost  a  gallon  ? 

OPERATION. 
18)42'75(2375  mills,  =  $2^375,  the  ansiver. 
36 

.      67  $  42^75  is  4275  cents.     After  bringing 

54  down  the  last  figure  in  the  dividend,  and 

dividing,  there  is  a  remainder  of  9  cents, 

which,  by  annexing  a  cipher,  is  reduced 

to  mills,  (90,)  in  which  the  divisor  is  con- 

90  tained  5  times,  which  is  5  mills,  and  there 

90  is  no  remainder.     Or,  we  might  have  re* 

^    duced  $  42*75  to  mills,  before  dividing,  by 

•  '*  annexing  a  cipher,  42750  mills,  which, 

divided  by  18,  would  have  given  the  same  result,  2375  mills, 
>vhich,  reduced  to  dollars,  is  $  2*375,  the  answer. 


135 
126 


66  DIVISION    OF    FEDERAL   MOBfUt;  If  SL 

4.  Divide  $  59^387  by  8. 

OPERATION. 

8)59'387 

Quotient j  7'423f ,  that  is,  7  dollars,  42  cents,  3  mills,  and  | 
of  another  mill.  The  f  is  the  remainder,  after  the  last  di- 
vision, written  over  the  divisor,  and  expresses  such  fractional 
part  of  another  mill.  For  all  purposes  of  business,  it  will  be 
sufficiently  exact  to  carry  the  quotient  only  to  mills,  as  the 
parts  of  a  mill  are  of  so  little  value  as  to  be  disregarded. 
Sometimes  the  sign  of  addition  (-J-)  is  annexed,  to  show  that 
there  is  a  remainder,  thus,  $  7'423  -[-. 

RUL.E. 

From  the  foregoing  examples,  it  appears,  that  division  of 
federal  money  does  not  diifer  from  division  of  simple  num- 
bers. The  quotient  will  he  the  answer  in  the  lowest  denomina- 
tionin  the  green  sur/ij  which  may  then  be  reducedto  dollars 

Note.  If  the  sum  to  be  divided  contain  only  dollars,  or 
dollars  and  cents,  it  may  be  reduced  to  mills,  by  annexing 
ciphers  before  dividing ;  or,  we  may  first  divide,  annexing 
ciphers  to  the  remainder,  if  there  shall  be  any,  till  it  shall 
be  reduced  to  mills,  and  the  result  will  be  the  same. 

exampl.es  for  practice. 

5.  If  I  pay  $  468'7o  for  750  pounds  of  wool,  what  is  the 
value  of  1  pound  ?  Ans,    $  0'625  ;  or  thus,  $  0^62^, 

6.  If  a  piece  of  elcth,  measuring  125  yards,  cost  $  181^25, 
v/hat  is  that  a  yard  ?  Ans.    $  1'45. 

7.  If  536  quintals  of  fish  cost  $  1913^52,  how  much  is  that 
a  quintal  ?  Am.    $  3^57. 

8.  Bought  a  farm,  containing  84  acies,  for  $3213  ;  what 
did  it  cost  me  per  acre  ?  Ajis.    $  38^25. 

9.  At  $  954  for  3816  yards  of  flannel,  what  is  that  a  vard  ? 

Ans.    $0'25. 

10.  Bought  72  pounds  of  raisins  for  $  8  ;  what  was  that 
a  pound  ?     ^,-  =:  how  much  ? 

Ans.    $0411^;  or,  $0411+. 

11.  Divide  $  12  into  200  equal  parts;  how  much  is  07?e 
of  the  parts  ?     ^^^^r  ~  liow  mnch  ?  Ans.    $  0*006. 

12.  Divide  $  30  by  750.     y^  ~  ^^^^''^  much  ? 

13.  Divide  $60  by  1200.     yf^Ty  =  how  much  ? 

14.  Divide  $215  into  86  equal  parts;  how  much  will 
one  of  the  parts  be  ?    -^g^-  ==  ^^^^"^  much  ? 


\ 


^   3^1.     SUPPLEMENT  TO  FEDERAL  MONEY.        6t 

15.  Divide  $  176  equally  among  250  men ;  how  much 
will  each  man  receive  ?     ^|4  z=.  how  much  ? 


SUPPLEMENT  TO  FEDERAL  MONEY. 

QUESTIONS. 

1.  What  is  understood    by  simple  numbers?      2.    *' 

by  compound  numbers?  3.  by  different  denomina- 
tions ?  4.  What  is  federal  money  ?  5.  What  are  the  de- 
nominations used  in  federal  money  ?  6.  How  are  dollars 
distinguished  from  cents  ?  7.  Why  are  two  places  assigned 
for  cents,  while  only  one  place  is  assigned  for  mills  ?  8. 
To  what  does  the  relative  value  of  mills,  cents,  and  dollars 
correspond  ?      9.  How  are  mills  reduced  to  dollars  ?     10. 

• to  cents?    11.    Why?      12.  How    are  dollars  reduced 

to   cents?      13.  to  mills?      14.  Why?     15.  How  is 

the    additioD  of    federal    money    performed?        16.    

subtraction?  17.  multiplication?  18.  divi- 
sion? 19.  Of  what  name  is  the  product  in  multiplication, 
and  the  quotient  in  division  ?  20.  In  case  dollars  only  are 
given  to  be  divided,*  what  is  to  be  done  ?  21.  When  is  one 
number  or  quantity  said  to  be  an  aliquot  part  of  another  ? 
22.  What  are  some  of  the  aliquot  parts  of  a  dollar  "^  23. 
When  the  price  is  an  aliquot  part  of  a  dollar,  how  may  the 
cost  be  found  ?  24.  What  is  this  manner  of  operating 
called  ?  25.  How  do  3^ou  find  the  cost  of  articles,  sold  by 
the  100  or  1000? 

EXERCISES. 

1.  Bought  23  firkins  of  butter,  each  containing  42  pounds, 
for  16^  cents  a  pound ;  what  would  that  be  a  firkin,  and 
how  much  for  the  whole  ?         Ans.    $  159'39  for  the  whole. 

2.  A  mon  killed  a  h^^i^  which  he  sold  as  follows,  viz.  the 
hind  quarters,  weighing  129  pounds  each,  for  5  cents  a 
pound  ;  the  fore  quarters,  one  weighing  123  pounds,  and  the 
other  125  pounds,  for  4J  cents  a  pound;  the  hide  and  tal- 
low, weighing  163  pounds,  for  7  cents  a  pound;  to  what 
did  the  whole  amount  ?  Ans,    ,$  35*47. 

3.  A  farmer  bought  25  pounds  of  clover  seed  at  11  cents 
a  pound,  3  pecks  of  herds  grass  seed  for  $  2^25,  a  barrel  of 
flour  for  $6'50,  13  pounds  of  sugar  at  12J  cents  a  pound; 
for  which  he  paid  3  cheeses,  each  vv^eighhig  27  pounds,  at 
8^  cents  a  pound,  and  5  barrels  of  cider  at  $  V25  a  barrel. 
The  balance  between  tlie  articles  bought  and  sold  is  1  cent 
is  it /or,  or  a^jainst  the  farmer  ? 


6S  SUPPLEMENT   TO   FEDERAL   MONEY.  If  32. 

4.  A  man  dies,  leaving  an  estate  of  $  71600 ;  there  are 
demands  against  the  estate,  amountirg  to  $  39876*74 ;  the 
residue  is  to  be  divided  between  7  sons;  what  will  each 
one  receive  ? 

5.  How  much  coffee,  at  25  cents  a  pound,  may  be  had  for 
100  bushels  of  rye,  at  87  cents  a  bushel  ?     Ans,  348  pounds. 

6.  At  12J  cents  a  pound,  what  must  be  paid  for  3  boxes 
of  sugar,  each  containing  126  pounds  ? 

7.  If  650  men  receive  $86'75  each,  what  will  they  all 
receive  ? 

8.  A  merchant  sold  275  pounds  of  iron,  at  6J  cents  a 
pound,  and  took  his  pay  in  oats,  at  $  0'50  a  bushel ;  how 
many  bushels  did  he  receive  ? 

9.  How  many  yards  of  cloth,  at  $  4*66  a  yard,  must  be 
given  for  18  barrels  of  flour,  at  $9*32  a  barrel  ? 

10.  What  is  the  price  of  three  pieces  of  cloth,  the  first 
containing  16  yards,  at  $  3'75  a  yard ;  the  second,  21  yards, 
at  $  4*50  a  yard  ;  and  the  third,  35  yards,  at  $  5*12^  a  yard  ? 

IT  32.  It  is  usual,  when  goods  are  sold,  for  the  seller  to 
deliver  to  the  buyer,  with  the  goods,  a  bill  of  the  articles 
and  their  prices,  with  the  amount  cast  up.  Such  bills  are 
sometimes  called  bills  of  parcels. 

Boston,  January  6, 1827. 
Mr.  Mel  Atlas 

Bought  ofBenj.  Burdett 

12^  yards   figured  Satin,  at  $  2*50  a  yard,  $31*2I» 

8    ...;...  sprigged  Tabby,...      1*25 10*00 

Received  payment,  $41*25 

Benj.  Burdett. 


Salem,  June  4, 1827. 
Mr.  James  Paywell 

Bought  of  Simeon  Thrifty 

S  hogsheads  new  Rum,  118  gal.  each,     at  $0*31    a  gal. 

2  pipes  French  Brandy,  126  and  132  gal.  ..     1*12  J 

1  hogshead  brown  Sugar,  Of  cwt.  ..  10*34    ..  cwt- 

3  casks  of  Rice,  269  lb.  each,  ..       *05    ..lb. 

5  bags  Coffee,  75  lb.  each,  .-     *.       *23    

1  chest  hyson  Tea,  861b.  ..       *92    


Received  payment,  $706*62j 

For  Simeon  Thrifty,  ^ 

Pkter  Faithful, 


^  32,  33.  REDUCTION.  69 

Wilderness,  February  8, 1827. 
Mr.  Peter  Carpenter 

(See  ir  30.)  Bought  of  Asa  Fdliree 

5682  feet  Boards,          at  $  6       per  M. 

2000 8'34 

800 Thick  Stuff,   ..    12^64 

1500 Lathing,  ..      4'      

650 Plank,  ..    10' 

879 Timber,          ..  2'50  .....  C. 

236 2^75 


Received  payment,  $  101^849 

Asa  Falltree. 
Note,     M.  stands  for  the  Latin  mille^  which  signifies  1000, 
and  C.for  the  Latin  word  centum^  which  signifies  100. 


IT  33.  We  have  seen,  that,  in  the  United  States,  money 
is  reckoned  in  dollars,  cents,  and  mills.  In  England,  it  is 
reckoned  in  pounds,  shillings,  pence,  and  farthings,  called 
denominations  of  money.  Time  is  reckoned  in  years,  months, 
weeks,  days,  hours,  minutes,  and  seconds,  called  denomina- 
tions of  time.  Distance  is  reckoned  in  miles,  rods,  feet,  and 
inches,  called  denominations  of  measure,  &:c. 

The  relative  value  of  these  denominations  is  exhibited  in 
tables,  which  the  pupil  miist  commit  to  memory. 


ENGLISH  MONEY. 


The  denominations  are  pounds,  shillings,  pence,  and  far- 
things. 

TABLE. 
4  farthings  (qrs.)  make  1  penny,   marked   d. 
12  pence      -    -    -    -     i  shilling,     -    -     s. 
20  shillings  -    -    -    -     1  pound,       -    -     £, 
Note,     Farthings  are  often  written  as  the  fraction  of  a 
penny ;  thus,  1  farthing  is  written  J  d.,  2  farthings,  ^  d.,  S 
farthings,  f  d. 


70  REDUCTION. 

How  many  farthings  in 
penny?    in  2    nt 


IT  35. 


1 
in   2    pence  ? 

— —  in  3  pence  ?    in  6 

pence  ? in  8  pence  ? 

in  9  pence  ? in  12  pence  ? 

•- in  1  shilling  ?  in  2 

shillings  ? 

How  many  pence  in  2  shil- 
lings ?   in  3  s.  ?   in 

4  s.  ?    in  6  s.  ?    in 

8  s.  ? in  10  s.  ? in  2 

shillings  and  2  pence  ?  


-  in  2  s.  4  d.  ? 
in  3  s.  6  d.  ? 


in  2  s.  3  d.  ? 
in  2  s.  6  d.  ? 
in  4  s.  3  d.  ? 

How  many  shillings   in   1 

pound  ?    in  2ie .  ?  

mZ£,?  in  4£ .  ?  

in  4iS .  6  s.  ? in  6^ .  8  s.  ? 

in   3£.   10  s.?  in 

2£.  15  s.? 


How  many  pence  in  4  far- 
things ?  in  8  farthings  ? 

in  12  farthings  ?  in 

24  farthings  ?  in  32  far- 


thing 


-  in  36  farthings  ; 


in  48  qrs.  ?    How  many 

shillings  in  48  qrs.  ?  in 

96  qrs.  ? 

How  many  shillings  in  24 

pence  ? in  3C  d.  ?  

in48d.  ?  in72d.? 

in  96  d.  ? in  120  d.  ? 

in  20  d.  ? in  27  d.  ? 

in  28  d.  ? in  30  d.  ? 

in  42  d.  ?  in  51  d.  ? 

How  many  pounds  in  20  shil- 

lings  ?  in  40  s.  ? in 

60  s.  ?  ■  in  80  s.  ? in 

86  s.  ? -in  128  s.  ? in 

70  s.  ?  in  55  s.  ? 


It  has  already  heen  remarked,  that  the  changing  of  ojie 
kind,  or  denomination,  into  another  kind,  or  denomination, 
without  altering  their  value,  is  called  Reduction.  (IF  27.) 
'J'hus,  when  we  change  shillings  into  pounds,  or  pounds  into 
shillings,  we  are  said  to  reduce  them.  From  the  foregoing 
examples,  it  is  evident,  that,  when  we  reduce  a  denomina- 
tion of  greater  value  into  a  denomination  of  less  value,  th« 
reduction  is  performed  by  multiplication  ;  and  it  is  then  call 
ed  Reduction  Descending,  But  when  we  reduce  a  denomina- 
tion of  less  value  into  one  of  greater  value,  the  reduction  is 
performed  by  division  ;  it  is  then  called  Reduction  Ascending, 
Thus,  to  reduce  pounds  to  shillings,  it  is  plain,  we  must 
multiply  by  20.  And  again,  to  reduoe  shillings  to  pounds, 
we  must  divide  by  20.  It  follows,  therefore,  that  reduction 
descending  and  ascending  reciprocally  prove  each  other. 


IT  33,  34. 


REDUCTION. 


71 


1.  Inl7ie.  13  s.  6Jd.  how 
many  farthings  ? 
OPERATION. 
£.     s.    d.  qrs 
17  13  6  3 
20  5. 

3535.  in  17ie.  13  5. 
\2d. 


4242  d. 

4^. 


16971  qrs.  the  Ans, 

In  the  above  example,  be- 
cause 20  shillings  make  1 
pound,  therefore  we  multiply 
\1£.  by  20,  increasing  the 
product  by  the  addition  of  the 
given  shillings,  (13,)  which, 
it  is  evident,  must  always  be 
done  in  like  cases ;  then,  be- 
cause 12  pence  make  1  shil- 
ling, we  multiply  the  shillings 
(353)  by  12,  adding  in  the 
given  pence,  (6.)  Lastly, 
because  4  farthings  make  1 
penny,  we  multiply  the  pence 
<'4242)  by  4,  adding  in  the 
given  farthings,  (3.)  We 
then  find,  that  in  17£.  13  s. 
6fd.,  are  contained  16971 
farthings. 

^  34.  The  process  in  the 
fully  examined,  will  be  found 

To  reduce  high  deTiomina- 
tions  to  lower, — Multiply  the 
highest  denomination  by  that 
number  which  it  takes  of  the 
next  less  to  make  1  of  this 
higher,  (increasing  the  pro- 
duct by  the  number  given, 
if  any,  of  that  less  denomina- 


2.  In  16971  farthings,  hoTT 
many  pounds .? 

OPERATION. 
Farthings  in  a  penny,  4)16971     3  qrs* 

Pence  in  a  shilling,      1 2  )4242    6  d- 

Shillings  in  a  pound,     2|0)35l3    13*. 

17^. 
Ans.  17  £.  135.  6fJ. 

Farthings  will  be  reduced 
to  pence,  if  we  divide  them 
by  4,  because  every  4  far- 
things make  1  penny.  There- 
fore, 16971  farthings  divided 
by  4,  the  quotient  is  4242 
pence,  and  a  remainder  of  3, 
which  is  farthings,  of  the 
same  name  as  the  dividend. 
We  then  divide  the  pence 
(4242)  by  12,  reducing  them 
to  shillings ;  and  the  shillings 
(353)  by  20,  reducing  them 
to  pounds.  The  last  quotient, 
17iS.,  with  the  several  re- 
mainders, 13  s.  6  d.  3  qrs.  con- 
stitute the  answer. 

Note.  In  dividing  353  s.  by 
20,  we  cut  oiF  the  cipher,  &c., 
as  taught  IT  22. 

foregoing  examples,  if  care- 
to  be  as  follows,  viz. 

To  reduce  low  denominations 
to  higher, — Divide  the  lowest 
denomination  given  by  that 
number  which  it  takes  of  the 
same  to  make  1  of  the  next 
higher.  Proceed  in  the  same 
manner  with  each  succeeding 
denomination,  until  you  hav« 


n 


HEDUCTIOlSr, 


IT  34. 


tion.)  Proceed  in  the  same 
Planner  with  each  succeeding 
denomination,  until  you  have 
brought  it  to  the  denomination 
required. 


brought  it  to  the  denomination 
required. 


EXAMPLES    FOR   PRACTICE. 


B.  Reduce   Z2£.  15  s.  8d. 

^o  farthings. 

5.  In  29  guineas,  at  28  s. 
^ach,  how  many  farthings  ? 

7.  Reduce  $  183,  at  6  s. 
each,  to  pence  ? 

9.  In  15  guineas,  how 
many  pounds  ? 


4.  Reduce  31 4*^2  farthings 
to  pounds. 

6.  In  38976  farthings,  how 
many  guineas  ? 
\8.  Reduce  11736  pence  tc 
dollars. 

10.  Reduce  21^2.  to  guin- 
eas. 


Note.  We  cannot  reduce  guineas  directly  to  pounds,  but 
we  may  reduce  the  guineas  to  shillings^  and  then  the  shil- 
lings to  pounds. 


TROY  WEIGHT. 

By  Troy  weight  are  weighed  gold,*  silver,  jewels,  and  all 
iquors.  The  denominations  are  pounds,  ounces,  penny- 
weights, and  grains. 

TABLE. 
24  grains  (grs.)     make  1  pennyweight,  marked     pwt. 
20  pennyweights  -     -     1  ounce,     -----     oz. 


12  ounces 


-     1  pound. 


lb. 


11.  Bought  a  silver  tank- 
ard, weighing  3  lb.  5  oz.,  pay- 
ing at  the  rate  of  $  I'OS  an 
ounce ;  what  did  it  cost  ? 

13.  Reduce  2101b.  Soz. 
12  pwt.  to  pennyweights. 

15.  In  71b.  11  oz.  3  pwt. 
9  grs.  of  silver,  how  many 
.grains  ? 


12.  Paid  $  44^28  for  a  sil- 
ver tankard,  at  the  rate  of 
$  1^08  an  ounce;  what  did  it 
weigh  ? 

14.  In  50572  pwt.  how 
many  pounds  ? 

16.  Reduce  456S1  grains 
to  pounds. 


*  The  fineness  of  gold  is  tried  by  fire,  and  is  recxctied  m  carats,  by  which  is 
understood  the  24th  part  of  any  quantity ;  if  it  lose  ncoiinff  in  the  trial,  it  is  said 
to  be  24  carats  finej  if  it  lose  2  carats,  it  is  then  22  c£,r"Etsiiiie,  which  is  the  stand- 
ard for  gokl 

Silver  wl*ch  abides  the  fire  without  loss  is  said  Ic  c«  12  ounces  filne.  Tlie 
standard  for  silver  coin  is  11  oz.  2  pwts.  of  fine  s.".^ei,  and  18  pwts.  of  copper 
HKjlted  tog^ether 


IT  34.  REDUCTION.  73 

APOTHECARIES'  WEIGHT. 

Apothecaries'  weight*  is  used  hy  apothecaries  and  physi- 
ciaus,  in  compounding  medicines.  The  denominations  ar© 
pounds,  ounces,  drams,  scruples,  and  grains. 

TABLE. 

20  grains,  (grs.)  make  1  scruple,  marked  9. 

3  scruples     -     -     -     1  dram,      -     -     -  3 . 

8  drams   -     -    -     -     1  ounce,    -     -    -  g . 

12  ounces  -    -    -    -     1  pound,   -    -    -  ib. 

17.  In9tb.  8§.  15.  29. 1     18.  Reduce    65799  grs.  to 

19  grs.,  how  many  grains.        [pounds. 

AVOIRDUPOIS   WEIGHT.t 

By  avoirdupois  weight  are  weighed  all  things  of  a  coarse 
and  drossy  nature,  as  tea,  sugar,  bread,  flour,  tallow,  hay, 
leather,  medicines,  (in  buying  and  selling,)  and  all  kinds 
of  metals,  except  goU  and  silver.  The  denominations  are 
tons,  hundreds,  quarters,  pounds,  ounces,  and  drams. 

TABLE. 

16  drams,    (drs.)      make  1  ounce,     -     marked    -  oz. 

16  ounces     -----  1  pound,     -----  lb. 

28  pounds     -----  1  quarter,    -----  qr. 

4  quarters   -----  1  hundred  v/eight,     -     -  cwt. 

20  hundred  weight      -     -  1  ton,     ------  T. 

Note  1.  In  this  kind  of  weight,  the  words  gross  and  net 
are  used.  Gross  is  the  weight  of  the  goods,  together  with 
the  box,  bale,  bag,  cask,  &c.,  which  contains  them.  Net 
weight  is  the  weight  of  the  goods  only,  after  deducting  the 
weight  of  the  box,  bale,  bag,  or  cask,  &c.,  and  all  other  al- 
lowances. 

Note  2.  A  hundred  weight,  it  will  be  perceived,  is  1 12  lb. 
Merchants  at  the  present  time,  in  our  principal  sea-ports, 
buy  and  sell  by  the  100  pounds. 

*  The  pouiic!  and  ounce  apcthecarics'  vvciglit,  and  the  pound  arid  ounce  Troy, 
are  the  same,  only  differonlly  divided,  and  suodiddcd. 

t  175  oz.  Troy  =2:  192  oz.  avoirdupois,  and  175  )b.  Troy -=  lit  lb.  avoirdu- 
DOJs.    1  lb.  Troy  =^  .5760  grdns,  and  1  )b.  avoirdupois  =s  7000  grains  Troy* 
G 


74 


REDUCTION. 


IF  34. 


19.  What  willScwt.  3  qrs. 
171b.  of  sugar  come  to,  at 
12J  cents  a  pound. 

21.  A  merchant  would  put 
109  cwt.  0  qrs.  12  lb.  of 
raisins  into  boxes,  containing 
26  lb.  each ;  how  many  boxes 
will  it  require  ? 

23.  In  12  tons,  15  cwt. 
1  qr.  191b.  6  oz.  12  dr.  how 
many  drams  ? 

25.  In  28  lb.  avoirdupois, 
how  many  pounds  Troy  ? 


20.  How  much  sugar,  at 
12 J-  cents  a  pound,  may  be 
bought  for  $  82*625  ? 

22.  In  470  boxes  of  raisins, 
containing  26  lb.  each,  how 
many  cwt.  ? 


24.  In  7323500  drams,  how 
many  tons  ? 

26.  In  34lb.  0  oz.  6  pwt. 
16  grs.  Troy,  how  many 
pounds  avoirdupois  ? 


CLOTK  MEASURE. 

Cloth  measure  is  used  in  selling  cloths  and  other  goods, 
sold  by  the  yard,  or  ell.  The  denominations  are  ells,  yards, 
quarters,  and  nails. 

TABILE, 

1  nails,  (na.)  or  9  inches,  make  1  quarter,     marked  qr. 

1  quarters,  or  36  inches,       -     1  yard,     -     -     -     -  yd. 

3  quarters,     -     -     -     -     -     -     1  ell  Flemish,     -     -  E.  Fl. 

5  quarters,     ------     1  ell  English,     -     -  E.  E. 

6  quarters,     ------     1  ell  French,      -     -  E.Fr. 


27.  In  573  yds.  1  qr.  1  na. 
now  many  nails? 

29.  In  151  ells  Eng.  how 
many  yards  ? 

Note.  Consult  ^  34,  ex.  3 . 


28.  In  9173  nails,  how  ma- 
ny yards  ? 

30.  In  188f  yards,  how  ma- 
ny ells  English  ? 


LONG  MEASURE. 

Long  measure  is  used  in  measuring  distances,  or  other 
things,  where  length  is  considered  without  regard  to  breadth. 
The  denominations  are  degrees,  leagues,  miles,  furlongs, 
rods,  yards,  feet,  inches,  and  barley-corns. 


IT  84; 


REDUCTION. 


75 


TABL.E. 

3  barley-corns,  (bar.)  make  1  incb,       -    marked    - 

1  foot,       -    -    -    -    - 


l2   inches, 
3   feet,    ------ 

5  J  yards,  or  16^  feet, 

40  rods,  or  220  yards,  -  - 
8  furlongs,  or  320  rods,  - 
3    miles,      -     -     -     -     - 

60    geographical,    or    69J 
statute  miles,  -     - 

360    degrees,       -    -    - 


1  yard,       -     -     -     -  - 

1  rod,  perch,  or  pole,  - 

1  furlong,       -     -    -  - 

1  mile,      -     -    -     -  - 

1  league,        -    -    -  - 


-\ 


1  degree,        -    - 

a  great   circle,   or 
ence  of  the  earth. 


m. 
ft. 
yd. 

fur. 
M. 
L. 

or  ^ 


deg. 
circumfer 


31.  How  many  barley-corns 
will  reach  round  the  globe,  it 
being  3G0  degrees  ? 

Note.  To  multiply  by  2,  is 
to  take  the  multiplicand  2 
times  ;  to  multiply  by  1,  is  to 
take  the  multiplicand  1  lime; 
to  multiply  by  j-,  is  to  take  the 
multiplicand  half  a  time,  that 
is,  the  half  of  it.  Therefore, 
to  reduce  360  degrees  to  stat- 
ute miles,  we  multiply  first  by 
the  whole  number,^  69,  and  to 
the  product  add  half  the  multi- 
plicand. Thus : 
J)  360 
69^ 

3240 
2160 

180  half  of  the  multiplicand. 

25020  statute  miles  in  360  de- 
grees. 

33.  How  many  inches  from 
Boston  to  the  city  of  Wash- 
ington, it  being  482  miles  ? 

35.  How  many  times  will  a 
wheel,  16  feet  and  6  inches 
in  circumference,  turn  round 
in  the  distance  from  Boston  to 
Providence,  it  being  40  miles  } 


32.  In  4755801600  barley- 
corns, how  many  degrees  ? 

Note.  The  barley-corns  be- 
ing divided  by  3,  and  that 
quotient  by  12,  we  have 
132105600  feet,  which  are  to 
be  reduced  to  rods.  We  can- 
not easily  divide  by  16J  on 
account  of  the  fraction  ^ ;  but 
16^  feet  =  33  half  feet^  in  1 
rod ;  and  132105600  feet  =z 
264211200  half  feety  which, 
divided  by  33,  gives  800G400 
rods. 

Hence,  when  the  divisor  is 
encumbered  with  a  fraction, 
^  or  J,  &c.,  we  may  reduce 
the  divisor  to  halves,  or  fourths, 
&c.,  and  reduce  the  dividend 
to  the  same;  then  the  quo- 
tient will  be  the  true  answer. 

34.  In  30539520  inched, 
how  many  miles  ? 

36.  If  a  wheel,  16  feet  6 
inches  in  circumference,  turn 
round  12800  times  in  going 
from  Boston  to  Providence^ 
what  is  the  distance  ? 


76 


REDUCTION. 


it  S6. 


LAND  OR  SQUARE  MEASURE. 

Square  measure  is  used  in  measuring  land,  and  any  other 
tiding,  where  length  and  breadth  are  considered.  The  de- 
nominations are  miles,  acres,  roods,  perches,  yards,  feet  and 
inches. 

IT  35,  3  feet  in  length  make  a  yard  in  long  measure  ;  but  it 
requires  3  feet  in  length  and  3  feet  in  breadth  to  make  a  yard 
in  square  measure ;  3  feet  in  length  and  one.  foot  wide  make 
3  square  feet ;  3  feet  in  length  and  2  feet  wide  make  2 
times  3,  that  is,  6  square  feet ;  3  feet  in  length  and  3  feet 
wide  make  3  times  3,  that  is,  9  square  feet.  This  will 
clearly  appear  from  the  annexed  figure. 

3  feet  =  1  yard. 


It  is  plain,  also,  that  a  square  foot, 
that  is,  a  square  12  inches  in  length 
and  12  inches  in  breadth,  must  con- 
tain 12  X  12  :=  X44  square  inches. 


11 

5 

o 


144 


TABIiE. 

square  inches  nz  12  X  12 ;  that  is,  ^ 
12  inches  in  length  and  12  inches  >  make  1  square  foot, 
in  breadth     ------      N 

9    square  feet  =;  3  X  3 ;  that  is,  3  feet 

in  length  and  3  feet  in  breadth 

30^  square  yards  —  5^  X  ^y  or  272^ 

square  feet  =  16^  X  16^, 
40    square  rods,        ------ 

4  roods,  or  160  square  rods, 

640  acres,      ------ 


1  square  yard. 

(  1  square  rod, 
'  (  perch  or  pole. 
.     1  rood. 

•  1  acre. 

•  1  square  mile. 

Note.  Gunter's  chain,  used  in  measuring  land,  is  4  rods 
in  length.  It  consists  of  100  links,  each  link  being  7-^ 
inches  in  length ;  25  links  make  1  rod,  long  measure,  and 
625  square  links  make  1  square  rod. 


!r  35,  36. 


REDUCTIOX. 


77 


37.  In  17  acres  3  roods  12 
rods,  how  many  square  feet  ? 

Note,  In  reducing  rods  to 
feet,  the  multiplier  will  be 
272  J.  To  multiply  by  J-,  is  to 
take  a  fourth  part  of  the  mul- 
tiplicand. The  principle  is 
the  same  as  shown  IT  34, 
ex.  31. 


39.  Reduce  64  square  miles 
o  square  feet  ? 

41.  There  is  a  town  6  miles 
square  ;  how  many  square 
miles  in  that  town  ?  how 
many  acres? 


^  38.  In  778457  square  feet, 
how  many  acres  ? 

Note.  Here  we  have  776457 
square  feet  to  be  divided  by 
2 72 J-.  Reduce  the  divisor  to 
fourths^  tliat  is,  to  the  lowest 
denomination  contained  in  it ; 
then  reduce  the  dividend  to 
fourths^  that  is,  to  the  same 
denomination,  as  shown  1132, 
ex.  34. 

40.  In  1,784,217,600  square 
feet,  how  many  square  miles  ? 

42.  Reduce  23040  acres  to 
square  miles. 


SOLID  OR  CUBIC  MEASURE. 

Solid  or  cubic  measure  is  used  in  measuring  things  that 
have  length,  breadth,  and  thickness;  such  as  timber,  w^ood, 
stone,  bales  of  goods,  &c.  The  denominations  are  cords, 
tons,  yards,  feet,  and  inches. 

IT  36.  It  has  been  shovv^n,  that  a  square  yard  contains 
3  X  3  =r,  9  square  feet.  A  cubic  yard  is  3  feet  long,  3  feet 
wide,  and  3  feet  thick.  Were  it  3  feet  long,  3  {^fti  wide, 
;  nd  one  foot  thick,  it  would  contain  9  cubic  feet ;  if  2  feet 
!iick,  it  would  contain  2  X  9  =  18  cubic  feet;  and,  as  it  is 
3  feet  thick,  it  does  contain  3  X  9  i=  27  cubic  feet.     This 

will  clearly  appear  from  the 
annexed  figure. 

It  is  plain,  also,  that  a  cubic 
foot,  that  is,  a  solid,  12  inches 
in  length,  12  inches  in  breadth, 
and  12  inches  in  thickness, 
will  contain  12  X  12  X  12  = 
1728  solid  or  cubic  inches. 


78  REDUCTION.  1[  S6^ 

TABLE. 

'« 

1728  solid  inehes,  z=  12  X  12  X  12,  ^ 

that  is,  12  inches  in  length,  >  make  1  solid  foot 
12  in  breadth,  12  in  thickness,  ) 
27  solid  feet,  :^  3  X  3  X  3     -    -    -     -     1  solid  yard. 
40  feet  of  round  timber,  or  50  feet  >  ,  ^  ,     , 

of  hewn  timber, I'    '     1  ton  or  load. 

128  solid  feet,  —  8  X  4  X  4,  that  ) 

is,  8  feet  in  length,  4  feet  in  >  -     -     1  cord  of  wood, 
width,  and  4  feet  in  height,    ) 

Note.  What  is  called  a  cord  foot ^  in  measuring  wood,  is 
16  solid  Uiei ;  that  is,  4  feet  in  length,  4  feet  in  width,  and 
1  foot  in  height;  and  8  such  feet,  that  is,  8  card  feet  make 
1  cord. 


44.  In  622080  cubic  inche5> 
how  many  tons  of  round  tim- 
ber ? 

46.  In  592  solid  feet  of 
wood,  liow  many  cord  feet  r 

48.  In  8  cards  of  wood,  how 
many  cord  feet  ? 

50.  2048  solid  feet  of  wood, 
how  many  cord  feet?  how  I  how  many  cord  feet?  how 
many  solid  feet?  jmany  cords? 


43,  Reduce  9  tons  of  round 
timber  to  cubic  inches. 

45.  In  37  cord  feet  of  wood, 
how  many  solid  feet  ? 

47.  Reduce  64  cord  feet  of 
wijcd  to  cords. 

49.   In  IG  cords  of  w^ood, 


WINE  MEASURE. 


Wine  measure  is  used  in  measuring  all  spirituous  liquors, 
ale  and  beer  excepted  ;  also  vinegar  and  oil.  The  denomi- 
nations  are  tuns^  pipes,  hogsheads,  barrels,  gallons,  quarts, 
pints,  and  gills. 

TABLE. 

4    gills  (gi.)        -     make  -     -     1  pint,      marked  pt. 

2    pints     -------1  quart,         -     -     -  qt. 

4    quarts        ------    ^  gallon,       -     -     -  gal. 

3H  gallons       ------     1  barrel,       -     -     -  bar. 

63"  gallons      ------     1  hogshead,       -     -  hhd. 

2    hogsheads      -----     i  pipe,  -     -     -  P. 

2    pipes,  or  4  hogsheids        -     1  tun,      -    -     -     -  T. 
Note,     A  gallon,  wine  measure,  contains  231  cubic  inches. 


TF  36.  REDUCTION.  79 


51.  Reduce  12  pipes  of  wine 
to  pints. 

53.  In  9  P.  1  hhd.  22  gals. 
3  qts.  how  many  gills  ? 

55.  In  a  tun  of  cider,  how 
many  gallons  ? 


52.  In  12096  pints  of  wine, 
how  many  pipes  ? 

54.  Reduce  39032  gills  to 
pipes. 

56.  Reduce  252  gallons  to 
tuns. 


ALE  OR  BEER  MEASURE. 

Ale  or  beer  measure  is  used  in  measuring  ale,  beer,  and 
milk.  The  denominations  are  hogsheads,  barrels,  gallons, 
quarts,  and  pints. 

TABLE. 

2  pints  (pts.)     -    make   -  1  quart,     -      marked  qt. 

4  quarts        -----  i  gallon,    -----  gal, 

36  gallons            -     -     -     -  i  barrel,    -----  bar. 

54  gallons      -----  i  hogshead,    -    -    -     -  hhd 

Note,     A  gallon,  beer  measure,  contains  282  cubic  inches. 


57.  Reduce  47  bar.  18  gal. 
of  ale  to  pints. 

59.  In  29  hhds.  of  beer, 
how  many  pints  ? 


58.  In  13680  pints  of  alC; 
how  many  barrels  ? 

60.  Reduce  12528  pints  to 
hogsheads. 


DRY  MEASURE. 


Dry  measure  is  used  in  measuring  all  dry  goods,  such  as 
grain,  fruit,  roots,  "salt,  coal,  &:c.  The  denominations  are 
chaldrons,  bushels,  pecks,  quarts,  and  pints. 

TABL,E. 

2  pints  (pts.)       make     -  1  quart,       -     marked     -  qt. 

8  quarts        -----  l  peck,        -----  pk. 

4  pecks        -----  1  bushel,      -----  bu. 

36  bushels      -----  1  chaldron,        -     -     -     -  ch , 

Note.     A  gallon,  dry  measure,  contains  268|- cubic  inches. 
A  Winchester  bushel  is  18^  inches  in  diameter,  8  inches 
deep,  and  contains  2150f  cubic  inches. 


80 


61.  In  75  bushels  of  wheat, 
how  many  pints  ? 

63.  Reduce  42  chaldrons  of 
coals  to  pecks. 


REDUCTION.  U  36,  37. 

62.  In  4800  pints,  how  ma- 
ny bushels  ? 

64.  In  6048  pecks,  how  ma- 
ny chaldrons  ? 


TIME. 

The  denominations  of  time  are  years,  months,  weeks, 
days,  hours,  minutes,  and  seconds. 

TABLE. 

60  seconds   (s.)    -      make      -     1  minute,      marked  m. 

60  minutes      ------     i  hour,      -     -     -  -  h. 

24  hours     -------i  day,        -     -     -  -  d. 

7  days      -------i  week,     -     -    -  -  w. 

4  weeks        ------     i  month,   -     -    -  -  mo 

13  months,  1  day  and  6  hours,  ^  1  common,  or  ) 

or  365  days  and  6  hours,   5     Julian  year,  5  "  ^^' 

IT  07,  The  year  is  also  divided  into  12  calendar  months, 
which,  in  the  order  of  their  succession,  are  numbered  as  fol- 
lows, viz, 

January,        1st  month,  has  31  days. 
February,     2d,      -     -     -     28 
March,  3d,      -     -     -     31 

Mayl'     ■  tt]  -"     "-     I  31  ^^^  ^^^'kT^-!^ 

June       -  6th 30  can  be  divided  by  4  with- 

July,       -  TO^ 31  outaremamder,itiscall. 

August,  8th  -    -     -  31  ed  leap  year,  m  which 

September,  9th, 30  February  has  29  days. 

October,  10th,  -     -     -  31 

November,  11th,  -    -    -  30 

December,  12th,  -    -    -  31 

The  number  of  days  in  each  month  may  be  easily  fixed  in 
the  mind  by  committing  to  memory  the  following  lines : 

Thirty  days  hath  September, 
April,  June,  and  November, 
February  twenty-eight  alone ; 
All  the  rest  have  thirty-one. 


ttSl 


REDUCTION. 


81 


The  first  seven  letters  of  the  alphabet,  A,  B,  C,  D,  E^fF,  G, 
are  used  to  mark  the  several  days  of  the  week,  and  they  are 
disposed  in  such  a  manner,  for  every  year,  that  the  letter  A 
shall  stand  for  the  1st  day  of  January,  B  for  the  2d,  &c.  In 
pursuance  of  this  order,  the  letter  which  shall  stand  for  Sun" 
dai/j  in  any  year,  is  called  the  Dominical  letter  for  that  year.  " 
The  Dominical  letter  being  known,  the  day  of  the  week 
on  which  each  month  comes  in  may  be  readily  calculated 
from  the  following  couplet : 

At  Dover  Dwells  George  Brown,  Esquire,  * 

Good  Carlos  Finch  And  David  Fryer. 

These  words  correspond  to  the  12  months  of  the  year,  and 
the  first  letter  in  each  word  marks  the  day  of  the  week  on 
which  each  corresponding  month  comes  in;  whence  any  other 
day  may  be  easily  found.  For  example,  let  it  be  required 
to  find  on  what  day  of  the  week  the  4th  day  of  July  falls,  in  the 
year  1827,  the  Dominical  letter  for  which  year  is  G.  Good 
answers  to  July ;  consequently,  July  comes  in  on  a  Sunday ; 
wherefjore  the  4th  day  of  July  falls  on  Wednesday. 

Note,  There  are  two  Dominical  letters  in  leap  years, 
one  for  January  and  February,  and  another  for  the  rest  of 
the  year. 

65.  Supposing  your  age  to      66.  Reduce  475047465  se- 
be    15 y.    19  d.    lib.    37 m.  conds  to  years. 
45  s.,  how  many  seconds  old 
are  you,  allowing  365  days  6 
hours  to  the  year  ? 

67.  How  many  minutes  from 
the  1st  day  of  January  to  the 
14th  day  of  August,  inclusive- 

ly? 

69.  How  many  minutes  from 
the  commencement  of  the  war 
between  America  and  Eng- 
land, April  19th,  1775,  to  the 
settlement  of  a  general  peace, 
which  took  place  Jan.  20th, 
17S3? 


68.  Reduce  325440  minutes 
to  days. 


70.  In   4079160   minutes, 
how  many  years  ? 


62  SUPPLEMENT    TO    REDUCTION.  IT  37* 

CIRCULAR  MEASURE,  OR  MOTION. 

Circular  measure  is  used  in  reckoning  latitude  and  longi- 
tude ;  alsd  in  computing  the  revolution  of  the  earth  and 
other  planets  round  the  sun.  The  denominations  are  circles, 
signs,  degrees,  minutes,  and  seconds. 

TABL.E. 

60  seconds  (^')     -    make     -  1  minute,    -    marked    -  ' 

60  minutes         -----  i  degree,     -----  o 

30  degrees         -----  i  sign,   ------  s. 

12  signs,  or  360  degrees,       -  1  circle  of  the  zodiac. 

Note,  Every  circle,  whether  great  or  sriiall,  is  divisible 
into  360  equal  parts,  called  degrees. 

71.  Reduce  9  s.  13^  25'  to!  72.  In  1020300",  how  many 
seconds.  |  degrees  ? 


The  following  are  denominations  of  things  not  included  in 
the  Tables  :— 

12  particular  things     -      make      -     1  dozen. 
12  dozen      --------     i  gross. 

12  gross,  or  144  dozen,     -    -     -    -     i  great  gross. 

Also, 
20  particular  things     -     make        -     1  score. 
6  points  make  1  line,   (  used  in  measuring  the  length  of 
12  lines      -    -    1  inch,  (      the  rods  of  clock  pendulums. 

4  inches  -    -   1  hand,  \  ""^f  ™  measuring  the  height  of 

'  (      noises. 

6  feet       -    -   1  fathom,  used  in  measuring  depths  at  sea. 
112  pounds     -    -     make     -    -     1  quintal  of  fish. 
24  sheets  of  paper    -    make  -     1  quire. 
20  quires   -------1  ream. 


SUPPX-EMENT  TO  KEDUCTZON. 

QUESTIONS. 

1.  What  is  reduction?  2.  Of  how  many  varieties  is  re- 
duction ?  3.  What  is  understood  by  different  denominations^ 
as  of  money,  weight,  measure,  &c.  ^    4.  How  are  high  de- 


IT  37.  SUPPLEMENT    TO    REDUCTION.  8S 

nominations  brought  into  lower  ?  5.  How  are  low  denomi- 
nations brought  into  higher  ?  6.  What  are  the  denomina- 
tions of  English  money  ?     7.  What  is  the  use  of  Troy  weight, 

and   what   are   the   denominations  ?      8.  avoirdupois 

weight  ?     the  denominations  ?     9.  What  distinction  do 

you  make  between  gross  and  we/ weight?  10.  What  dis- 
tinctions do  you  make  between  long,  square,  and  cubic 
•neasure  ?  11.  What  are  the  denominations  in  longmea-, 
sure  ?  12. —  in  square  measure  ?  13.  — —  in  cubic  mea- 
sure ?  14.  How  do  you  multiply  by  ^  ?  15.  When  the  di- 
visor contains  a  fraction,  how  do  you  proceed  ?  16.  How  is 
the  superficial  contents  of  a  square  figure  found  ?  17.  How 
is  the  solid  contents  of  any  body  found  in  cubic  measure  ? 

18.  How  many  solid  or  cubic  feet  of  v/ood  make  a  cord  ? 

19.  What  is  understood  by  a  cord  foot?  20.  How  many 
such  feet  make  a  cord  ?     21.  What  are  the  denominations 

of  dry  measure  ?     22.  of  wine  measure  ?     23.  of 

time  ?  24.  of  circular  measure  ?  25.  For  what  is  cir- 
cular measure  used  ?  26.  How  many  rods  in  length  is  Gun- 
ter^s  chain  ?  of  how  many  links  does  it  consist  ?  how  many 
links  make  a  rod  ?  27.  How  many  rods  in  a  mile  ?  28.  How 
many  square  rods  in  an  acre  ?  29.  How  many  pounds  make 
1  cwt.  ? 

EXERCISES. 

1.  In  46i£ .  4  s.,  how  many  dollars  ?  Ans.    $  154. 

2.  In  36  guineas,  how  many  crowns,  at  6  s.  7d.  each  ? 

Ans.  153  crowns,  and  9d. 

3.  How  many  rings,  each  weighing  5  pwt.  7  grs.,  may  be 
made  of  31b.  5  oz.  16  pwt.  2  grs.  of  gold?  Aiis,  158. 

4.  Suppose  West  Boston  bridge  to  be  212  rods  in  length, 
how  many  times  will  a  chaise  wheel,  18  feet  6  inches  in 
circumference,  turn  round  in  passing  over  it  ? 

**  Ans,  1892^^2  times. 

5.  In  470  boxes  of  sugar,  each  26  lb.,  how  many  cwt.  ? 

6.  In  10  lb.  of  silver,  how  many  spoons,  each  weighing 
.    oz.  10  pwt.  ? 

7.  How  many  shingles,  each  covering  a  space  4  inches 
one  way  and  6  inches  the  other,  would  it  take  to  cover  1 
square  foot  ?;>  How  many  to  cover  a  roof  40  feet  long,  and 
24  feet  wide  ?     (See  If  25.)     Ans.  to  the  last^  5760  shingles, 

8.  How  many  cords  of  wood  in  a  pile  23  feet  long,  4  feet 
wide,  and  6  feet  high  ?  Ans.  4  cords,  and  7  cord  feet. 


84  SUPPLEMENT    TO    REDUCTION.  IT  87. 

9.  There  is  a  room  18  feet  iu  length,  16  feet  in  width, 
•iU>d  8  feet  in  height;  how  many  rolls  of  paper,  2  feet  wide, 
and  containing  1 1  yards  in  each  roll,  v/ili  it  take  to  cover  the 
v/alls?  Am,  8^. 

10.  How  many  cord  feet  in  a  load  of  wood  6^  feet  long, 

2  feet  wide,  and  5  feet  liigh  ?  Ans.  4y^g^  cord  feet. 

11.  If  a  ship  sail  7  miles  an  hour,  how  far  will  she  sail. 
£t  that  rate,  in  3  w.  4  d.  16  h.  ? 

12.  A  merchant  sold  12  hhds.  of  brandy,  at  §2'75  a  gal- 
lon ;  how  much  did  each  hogshead  come  to,  and  to  how 
much  did  the  whole  amount  ? 

13.  How  much  cloth,  at  7  s.  a  yard,  may  be  bought  for 
29£.  Is.? 

14.  A  goldsmith  sold  a  tankard  for  10£.  Ss.  at  the  rate 
of  5  s.  4  d.  per  ounce ;  how  much  did  it  weigh  ? 

15.  An  ingot  of  gold  weighs  2  lb.  8  oz.  16  pwt. ;  how 
much  is  it  worth  at  3  d.  per  pwt.  ? 

16.  At  $  048  a  pound,  what  will  1  T.  2  cwt.  3  qrs.  16  lb. 
of  lead  come  to  ? 

17.  Reduce  14445  ells  Flemish  to  ells  English. 

18.  There  is  a  house,  the  roof  of  which  is  44J  feet  in 
length,  and  20  feet  in  width,  on  each  of  the  two  sides;  if 

3  shingles  in  width  cover  one  foot  in  length,  how  many 
shingles  will  it  take  to  lay  one  course  on  this  roof?  if  3 
courses  make  one  foot,  Iiow  many  courses  will  there  be  on 
une  side  of  the  roof?  how  many  shingles  will  it  take  to 
cover  one  side  ? to  cover  both  sides  ? 

A.}is.   16020  shingles. 

19.  How  many  steps,  of  30  inches  each,  must  a  man  take 
in  travelling  54.j~  miles  ? 

20.  How  many  seconds  of  lime  would  a  person  redeem 
in  40  years,  by  rising  each  morning  j  hour  earlier  than  he 
now  does  ? 

"21.  If  a  man  lay  up  4  shillings  each  day,  Sundays  ex- 
cepted, how  many  dollars  would  he  lay  up  in  45  years  ? 

22.  If  9  candles  are  made  from  1  pound  of  tallow^  how 
many  dozen  can  be  made  from  24  pounds  and  10  ounces  ? 

23.  If  one  pound  of  wool  make  60  knots  of  yarn,  how 
many  skeins,  of  ten  knots  each,  may  be  spun  from  4  pounds 
6  ounces  of  wool  ? 


IT  38.  ADDITION   OF   COMPOUND  NUMBERS.  85 

ADSXTZOKT 

OF  COMPOUND  NUMBERS. 

M  38.  1.  A  boy  bought  a  knife  for  9  pence,  and  a  comb 
for  3  pence ;  how  much  did  he  give  for  both  ?  Ans.  1  shilling. 

2.  A  boy  gave  2  s.  6  d.  for  a  slate,  and  4  s.  6  d.  for  a  book ; 
how  much  did  he  give  for  both  ? 

3.  Bought  one  book  for  1  s.  6  d.,  another  for  2  s.  3  d.,  an- 
other for  7  d. ;  how  much  did  they  all  cost  ?       Ans,  4  s.  4  d. 

4.  How  many  gallons  are  2  qts.  +  3  qts.  +  1  qt.  ? 

6.  How  many  gallons  are  3  qts.  +  2  qts.  +  1  qt.  +  3 
qts.  +  2  qts.  ? 

6.  How  many  shillings  are  2  d.  +  3  d.  +  5  d.  -|-  6  d.  -[-  7  d.  ? 

7.  How  many  pence  are  1  qr.  -j-  2  qrs.  -j-  3  qrs.  +  2  qrs. 

+ 1  qr.  ^ 

8.  How  many  pounds  are  4  s.  -)-  10  s.  +  15  s.  +  1  s.  ?  * 

9.  How  many  minutes  are  30  sec.  -|-  45  sec.  -f"  20  sec.  ? 

10.  How  many  hours  are  40  min.  -f-  25  min.  -f-  6  min.  ? 

11.  How  many  days  are  4  h.  +  8  h.  +  10  h.  -j-  20  h.  ? 

12.  How  many  yards  in  length  are  I  L  -{-  2  L  -{-  I  L? 

13.  How  many  feet  are  4  in.  +  8  in.  -]-  10  in.  -|-  2  in. 
+  1  in.  ? 

14.  How  much  is  the  amount  of  1  yd.  2  ft.  6  in.  +  2  yds. 
1ft.  8in.? 

15.  What  is  the  amount  of  2  s.  6  d.  +4s.  3d.  +  7s.  8d.? 

16.  A  man  has  two  bottles,  which  he  wishes  to  fill  witli 
wine ;  one  will  contain  2  gal.  3  qts.  1  pt.,  and  the  other  3 
qts.  ;  how  much  wine  can  he  put  in  them  ? 

17.  A  man  bought  a  horse  for  15iB.  14  s.  6  d.,  a  pair  of 
oxen  for  20iS .  2  s.  8  d.,  and  a  cow  for  5£ .  6  s.  4  d. ;  what 
did  he  pay  for  all  ? 

When  the  numbers  a^e  large,  it  will  be  most  convenient 
to  write  them  down,  placing  those  of  the  same  kind,  or  de- 
nomination, directly  under  each  other,  and,  beginning  with 
those  of  the  least  value,  to  add  up  each  kind  separately.  ^ 
DERATION.  In  this   example,  adding  up  the 

column  of  pence,  we.find  the  amount 
to  be  18  pence,  which  being  =  1  s. 
6  d.,  it  is  plain,  that  we  may  write 

dpwn  the  6  d.  under  the  column  o^ 

Aas,  41       3     6  pence,  and  reserve  the  1  s.  to  be  add- 

ed in  with  the  other  shillings. 


£.  s.  d. 

15  14  6 

20  2  8 

5  6  4 


H 


86  ADDITION    OF    COMPOUND    NUMBERS.  V  38. 

Next,  adding  up  the  column  of  shillings,  together  with 
the  1  s.  which  we  reserved,  we  find  the  amount  to  be  23  s. 
zn  l£.  3  s.  Setting  the  3 s.  under  its  own  column,  we  add 
the  1  £ .  with  the  other  pounds,  and,  finding  the  amount  to  be 
UiS .,  we  write  it  down,  and  the  work  is  done. 

Am.  41iB.  3  s.  6  d. 

Note.  It  will  be  recollected,  that,  to  reduce  a  lower  into 
a  higher  denomination,  we  divide  by  the  number  which  it 
takes  of  the  lower  to  make  one  of  the  higher  denomination. 
In  addition,  this  is  usually  called  carrying  for  that  number : 
thus,  between  pence  and  shillings,  we  carry  for  12,  and  be- 
tween shillings  and  pounds,  for  20,  &c. 

The  above  process  may  be  given  in  the  form  of  a  general 
Rule  for  the  Addition  of  Compound  Numbers : 

I.  Write  the  numbers  to  be  added  so  that  those  of  the 
ame  denomination  may  stand  directly  under  each  other. 

II.  Add  together  the  numbers  in  the  column  of  the  lowest 
lenomination,  and  carry  for  that  number  which  it  takes  of 
he  same  to  make  one  of  the  next  higher  denomination, 
j^roceed  in  this  manner  with  all  the  denominations,  till  you 
ome  to  the  last,  whose  amount  is  written  as  in  simple  num- 
bers. 

Proof     The  same  as  in  addition  of  simple  numbers. 

EXAMPIiES   FOR   PRACTICE. 


£.       s.     d.     gr. 

46      11     3     2 

16        7     4     0 

538     19     7     1 

£.         s.      d. 
72        9       6J- 
18       0     10^ 
36      16        5f 

£. 
183 

8 

s. 
19 
17 
15 

d. 

4 

10 

4 

if),     oz.   jjwt.    gr 
36     7     10     11 
42     6        9     13 
81     7     16     15 

Troy  WeicIht. 

oz.   piot.  gr. 
6      14       9 
8       6     16 
3     11      10 

oz. 
3 

pwt. 

13 

7 

J 

18 

16 

4 

Bought  a  silver  tankard,  weighing  2  lb.  3  oz.,  a  silver 
cup,  weighing  3  oz.  10  pv/t.,  and  a  silver  thimble,  weighing 
2  p^vt.  13  grs. ;  what  was  the  weight  of  the  whole  ? 


38.             ADDITION    OF    COMPOUND    NUMBERS.  St 

) 

Ayoirdupois  Weight. 

r.     ciDt,  qr,    lb.      oz.      dr.                    ctot.   qr.     lb.  oz.  dr. 

14     11     1     16       5     10                  16     3     18  6  14 

25       0     2     11       9     15               i-         2     16  8  12 

7     18     0     25     11       9                                 22  11  10 


A  man  bought  5  loads  of  hay,  weighing  as  follows,  viz. 
23  cwt.  (  =  1  T.  3  cwt.)  2  qrs.  171b. ;  21  cwt.  1  qr.  16  lb. ; 
19cwt.  0  qr.  24  lb. ;  24  cwt.  3  qrs. ;  11  cwt.  Oqr.  lib.; 
how  many  tons  in  the  whole  ? 

Cloth  Measure. 

yds.   qr.  na.  E.  Fl.  qr.  na.  E.  En.  qr.  na. 

36     1     2  41     1     2  75     4     2 

41     2     3  18     2     3  31     1     0 

'     65     3     1  67     0     1  28     3     1 


There  are  four  pieces  of  cloth,  which  measure  as  follows, 
viz.  36  yds.  2  qrs.  1  na. ;  18  yds.  1  qr.  2  na. ;  46  yds.  3  qrs. 
3  na. ;  12  yds.  0  qr.  2  na. ;  how  many  yards  in  the  whole  ? 

Long  Measure. 

Deg.  mi.  fur.    r.  ft.  in.  bar. 

59  46  6  29  15  10  2 

216  39  1  36  14  6  1 

678  53  7  24  9  8  1 


Mi. 

fur 

.poL 

3 

7 

8 

6 

27 

Land  or  Square  Measure. 

Pol.     ft.        in.                            A.     rood  pol.      ft.  in. 

36  179  137          56  3  37  245  228 

19  248  119          29  1  28   93  25 

12   96   75         416  2  31  128  119 


88  ADDITION    OF    COMPOUND   NUMBERS.  IT  38. 

There  are  3  fields,  which  measure  as  follows,  viz.  17  A. 
3r.  16  p.;  28  A.  5  r.  18  p.;  11  A.  Or.  25  p.;  how  much 
laud  in  the  three  fields  ? 

Solid  or  Cubic  Measure. 


Ton.  ft. 

in. 

yds. 

ft. 

in. 

cords,    ft. 

29  36 

1229 

75 

22 

1412 

37  119 

12  19 

64 

9 

26 

195 

9  110 

8   11 

917 

3 

19 

1091 

48  127 

Wine  Measure. 

Hlid.  gal.  qts.  pts.  Tun.  hhd.  gal.  qts. 

51  53     1     1  37  2     37     2 

27  39     3     0  19  1     59     1 

9  13     0     1  28  2       0     0 


A  merchant  bought  two  casks  of  brandy,  containing  as 
follows,  viz.  70  gal.  3  qts. ;  67  gal.  1  qt. ;  how  many  hogs- 
heads, of  63  gal.  each,  in  the  whole  ? 

Dry  Measure. 

Bus.  ;?.     qt.   pt.  Ch.     bus.    p.   qts. 

36     2     5     1  48     27     3     5 

19     3     7     0  6     29     1     7 


Time. 

Y.      mo.    ID.     d.      h.      772.        5.  Y.  m.  w.  d. 

57     11     3     6     23     55     11  40  3     1  5 

84       9     2     0     16     42     18  16  7     0  4 

32       6     0     5       5     18       5  27  5     2  0 


H  39,  SUBTRACTION    OF    COMPOUND   NUMBERS.  89 

SUBTRACTION 

OF  COMPOUND  NUMBERS. 

^  39.  1.  A  boy  bought  a  knife  for  9  cents,  and  sold  it 
for  17  cents  ;  how  much  did  he  gain  by  the  bargain  ? 

2.  A  boy  bought  a  slate  for  2  s.  6  d.,  and  a  book  for  3  s.  6  d. ; 
how  much  more  was  the  cost  of  the  book  than  of  the  slate  ? 

3.  A  boy  owed  his  playmate  2  s. ;  he  paid  him  1  s.  6  d. ; 
how  much  did  he  then  owe  him  ? 

4.  Bought  two  books ;  the  price  of  one  was  4  s.  6  d.,  the 
price  of  the  other  3  s.  9  d. ;  what  was  the  difference  of  their 
costs  ? 

5.  A  boy  lent  5  s.  3d.;  he  received  in  payment  2  s.  6  d. ; 
how  much  was  then  due  ? 

6.  A  man  has  a  bottle  of  wine  containing  2  gallons  and  3 
quarts  ;  after  turning  out  3  quarts,  how  much  remained  ? 

7.  How  much  is  4  gal.  less  3  gal.  ?  4  gal.  —  (less)  2  qts.  ? 
4  gal.  —  1  qt.  ?  4  gal.  —  1  gal.  1  qt.  ?  4  gal.  —  1  gal.  2  qts.  ? 
4  gal.  —  1  gal.  3  qts.  ?  4  gal.  —  2  gal.  3  qts. .?    4  gal.   1  qt. 

—  1  gal.  3  qts.  ? 

8.  How  much  is  1  ft.  —  (less)  6  in.  ?  1  ft.  —  8  in.  ?  6  ft. 
3in.  — .  1ft.  6in.  ?  7ft.  8  in. —4ft.  2 in.  ?  7 ft.  8 in. —■  5  ft. 
10  in.  ? 

9.  What  is  the  difference  between  4£,  6  s.  and  1^.  8  s.? 

10.  How  much  is  3£.  —  (less)  Is.?  3^.  —  2  s.  ?  3^^. 

—  3s.?  3£,  —  15  s.?  Z£.  4s.  — 2ie.  6s.?  10i£.  4s.— 
5^.8.  s? 

11.  A  man  bought  a  horse  for  30iS.  4  s.  8d.,  and  a  cow 
for  5£ .  14  s.  6  d. ;  what  is  the  difference  of  their  costs  ? 

OPERATION.  Xs  the  two  numbers  are  large. 

Minuend       30       4     R         ^  ^^^^^  ^^  convenient  to   write 
Subtrahmd,     5     14    6         them  down,  the'  less  under  the 

' greater,  pence  under  pence,  shil- 

Ans.  24  10  2  lings  under  shillings,  &c.  We 
may  now  take  6  d.  from  8  d.,  and 
there  will  remain  2  d.  Proceeding  to  the  shillings,  we  can- 
not take  14  s.  from  4  s.,  but  we  may  borrow,  as  in  simple  num- 
bers, 1  from  the  pounds,  zz=  20  s.,  which  joined  to  the  4  s. 
makes  24  s.,  from  which  taking  14  s.  leaves  10  s.,  which  we 
set  down.  We  must  now  carry  1  to  the  5^.,  making  6£.^ 
which  taken  from  3Q,£ .  leaves  24iS.,  and  the  work  is  done. 
Note,     The  most  convenient  way  In  borrowing  is,  to  sub* 


90  SUBTRACTION    OF   COMPOUND   NUMBERS.        ft  S9, 

tract  the  subtrahend  from  the  figure  borrowed,  and  add  the 
d'ifFerence  to  the  minuend.  Thus,  in  the  above  example,  14 
from  20  leaves  6,  and  4  is  10. 

The  process  in  the  foregoing  example  may  be  presented 
in  the  form  of  a  Rule  for  the  Subtraction  of  Compound  Num- 
bers : 

I.  Write  down  the  sums  or  quantities,  the  less  under  the 
greater,  placing  those  numbers  which  are  of  the  same  de- 
nomination directly  under  each  other. 

II.  Beginning  with  the  least  denomination,  take  succes- 
sively the  lower  number  in  each  denomination  from  the  up- 
per, and  write  the  remainder  underneath,  as  in  subtraction 
of  simple  numbers. 

III.  If  the  lower  number  of  any  denomination  be  greater 
than  the  upper,  borrow  as  many  units  as  make  one  of  the 
next  higher  denomination,  subtract  the  lower  number  there- 
from, and  to  the  remainder  add  the  upper  number,  remem- 
bering always  to  add  1  to  the  next  higher  denomination  for 
that  which  you  borrowed. 

Proof  Add  the  remainder  and  the  subtrahend  together, 
as  in  subtraction  of  simple  numbers  ;  if  the  work  be  right, 
the  amount  will  be  equal  to  the  minuend. 

exampl.es  for  practice. 

1.  a  merchant  sold  goodsto  the  amount  of  136iS .  7  s.  6^d., 
and  received  in  payment  50^.  10  s.  4fd;  how  much  re- 
mained due  ?  Ans.  S5£,  17  s.  If  d. 

2.  A  man  bought  a  farm  for  1256dB.  10  s.,  and,  in  selling 
it,  lost  87iB .  10  s.  6  d. ;  how  much  did  he  sell  it  for  ? 

Ans.  U6S£,  19  s.  6  d. 

3.  A.  man  bought  a  horse  for  27 £ .  and  a  pair  of  oxen  for 
19<£ .  12  s.  8^  d. ;  how  much  was  the  horse  valued  more  than 
the  oxen  ? 

4.  A  merchant  drew  from  a  hogshead  of  molasses,  at  one 
time,  13  gal.  3  qts.  ;  at  another  time,  5  gal.  2  qts.  1  pt.  ; 
what  quantity  was  there  left  ?  Ans,  43  gal.  2  qts.  1  pt. 

5.  A  pipe  of  brandy,  containing  118  gal.  sprang  a  leak, 
when  it  was  found  only  97  gal.  3  qts.  1  pt.  ren^ained  in  the 
cask;  how  much  was  the  leakage ? 

6.  There  was  a  silver  tankard  which  weighed  3  lb.  4  oz. ; 
the  lid  alone  weighed  5  oz.  7pwt.  13  ^rs.;  how  much  did 
the  tankard  weigh  without  the  lid .? 


if  Sfd.         SUBTRACTION    OF    COMPOUND    NUMBERS.  91 

7.  From  15  lb.  2  oz,  5  pwt.  take  9  oz.  8  pwt.  10  grs. 

8.  Bought  a  hogshead  of  sugar,  weighing  9  cwt.  2  qrs. 
17  lb. ;  sold  at  three  several  times  as  follows,  viz.  2  cwt.  1  qr. 
11  lb.  5  oz. ;  2  qrs.  18  lb.  10  oz. ;  25  lb.  6  oz. ;  what  was  the 
weight  of  sugar  which  remained  unsold  ? 

Alls.  6  cwt.  1  qr.  171b.  11  oz. 

9.  Bought  a  piece  of  black  broadcloth,  containing  36  yds. 
2  qrs. ;  two  pieces  of  blue,  one  containing  10  yds.  3  qrs. 
2  na.,  the  other,  18  yds.  3  qrs.  3  na.  j  how  much  more  was 
there  of  the  black  than  of  the  blue  ? 

10.  From  28  miles,  5  fur.  16  r.  take  15  m.  6  fur.  26  r.  12  ft. 

11.  A  farmer  has  two  mowing  fields;  one  containing  13 
acres  6  roods;  the  other,  14  acres  3  roods:  he  has  two 
pastures  also;  one  containing  26  A.  2  r.  27  p. ;  the  other, 
45  A.  5  r.  33  p. :  how  much  more  has  he  of  pasture  than  of 
mowing  ? 

12.  From  64  A.  2  r.  11  p.  29  ft.  take  26  A.  5  r.  34  p.  132  ft. 

13.  From  a  pile  of  wood,  containing  21  cords,  was  sold,  at 
one  time,  8  cords  76  cubic  feet ;  at  another  time,  5  cords  7 
cord  feet ;  what  was  the  quantity  of  wood  left  ? 

14.  How  many  da^'s,  hours  and  minutes  of  any  year  will 
be  future  time  on  the  4th  day  of  July,  20  minutes  past  3 
o'clock,  P.  M.  ?  Ans.   180  days,  8  hours.  40  minutes. 

15.  On  the  same  day,  hour  and  minute  of  July,  given  in 
the  above  example,  what  will  be  the  difference  between  the 
past  and  future  time  of  that  month  ? 

16.  A  note,  bearing  date  Dec.  28th,  1826,  was  paid  Jan. 
2d,  1827  ;  how  long  was  it  at  interest  ? 

llie  distance  of  time  from  one  date  to  that  of  another  may 
be  found  by  subtracting  the  first  date  from  the  last,  observing 
to  number  the  months  according  to  their  order,  (^f  37.) 

OPERATION. 
*    jy  ^1827.       Istm.       2d  day.  Note,  In  casting  in- 

•  (  1826.     12 28  terest,  each  month  is 

Ans.  ~0        0         .      Td^  reckoned  30  days. 

17.  A  note,  bearing  date  Oct.  20th,  1823,  was  paid  April 
25th,  1825 ;  how  long  was  the  note  at  interest  ? 

18.  What  is  the  difference  of  time  from  Sept.  29,  1816,  to 
April  2d,  1819  ?  Ans,  2  v.  6  m.  3  d. 

19.  London  is  51°  32',  and  Boston  42°  23'  N.  latitude; 
what  is  the  difference  of  latitude  between  the  two  places  ? 

Am.  9°  9', 


92  SUBTRACTION    OF    COMPOUND   NUMBERS.         IT  40. 

20.  Boston  is  71°  3',  and  the  city  of  Washington  is  77^ 
43'  W.  longitude ;  what  is  the  difference  of  longitude  be- 
tween the  two  places  ?  Am.  6°  40'. 

21.  The  island  of  Cuba  lies  between  74^  and  85°  W.  lon- 
gitude} how  many  degrees  in  longitude  does  it  extend? 

IT  40.  1.  When  it  is  12  o'clock  at  the  most  easterly  ex- 
tremity of  the  island  of  Cuba,  what  will  be  the  hour  at  the 
most  westerly  extremity,  the  difference  in  longitude  be- 
ing 11°? 

Note.  The  circumference  of  the  earth  being  360°,  and 
the  earth  performing  one  entire  revolution  in  24  hours,  it 
follows,  that  the  motion  of  the  earth,  on  its  surface,  from 
west  to  east,  is 

15°  of  motion  in  1  hour  of  time;  consequently, 
1°  of  motion  in  4  minutes  of  time,  and 
1'  of  motion  in  4  seconds  of  time. 

From  these  premises  it  follows,  that,  when  there  is  a  dif- 
ference in  longitude  between  two  places,  there  will  be  a 
corresponding  difference  in  the  hour,  or  time  of  the  day. 
The  difference  in  longitude  being  15°,  the  difference  in  time 
Avill  be  1  hour,  the  place  easterly  having  the  time  of  the  day 
1  hour  earlier  than  the  place  ivesterly,  which  must  be  par- 
ticularly regarded. 

If  the  difference  in  longitude  be  1°,  the  difference  in  time 
will  be  4  minutes,  &c. 

Hence, — If  the  difference  in  longitude,  in  degrees  and 
minutes,  between  two  places,  be  multiplied  by  4,  the  pro- 
duct will  be  the  difference  in  time,  in  minutes  and  seconds, 
which  may  be  reduced  to  hours. 

We  are  now  prepared  to  answer  the  above  question. 

11°  Hence,  when  it  is  12  o'clock  at  tiie 

4  most  easterly  extremity  of  the  island,, 

"" —     .  it  will  be  16  minutes  past  11  o'clock 

44  minutes.         ^^  ^j^^  ^^^^^  western  extremity. 

2.  Boston  being  6°  40'  E.  longitude  from  the  city  of 
Washington,  when  it  is  3  o'clock  at  the  city  of  Washington, 
what  is  the  hour  at  Boston  ? 

Ans.  26  minutes  40  seconds  past  3  o'clock. 

3.  Massachusetts  being  about  72°,  and  the  Sandwich 
Islands  about  155°  W.  longitude,  when  it  is  28  minutes  past 
6  o'clock,  A.  M.  at  the  Sandwich  Islands,  what  will  be  the 
liour  in  Massachusetts  ?  Ans.  12  o'clock  at  noon* 


^B"  41*      MULTIPLICATION  OF  COMPOUND  NUMBERS. 


93 


MUKTXPKICATION   &   DIVISZOZff 

OF  COMPOUND  NUMBERS. 


IT  41.  1.  A  man  bought  2  yards  of  cloth,  at  1  s.  6  d.  per 
yard ;  what  was  the  cost  ? 

2.  If  2  yards  of  cloth  cost  3  shillings,  what  is  that  per 
yard  ? 

3.  A  man  has  three  pieces  of  cloth,  each  measuring  10 
yds.  3  qrs. ;  how  many  yards  in  the  whole  ? 

4.  If  3  equal  pieces  of  cloth  contain  32  yds.  1  qr.,  how 
much  does  each  piece  contain  ? 

5.  A  man  has  five  bottles,  each  containing  2  gal.  1  qt.  1  pt. ; 
how  much  wine  do  they  all  contain  ? 

6.  A  man  has  11  gal.  3  qts.  1  pt.  of  wine,  which  he  would 
divide  equally  into  five  bottles  ;  how  much  must  he  put  into 
each  bottle  ? 

7.  How  many  shillings  are  3  times  8  d.  ?  3  X  9  d.  ? 

> 3  X  lOd.?  . 4X  ^d.  ?  7X6d.?  10  X 

9  d.  ?  2X3  qrs.  ?  5X2  qrs.  ? 

8.  How  much  is  one  third  of  2  shillings  ? ^  of  2  s. 

3  d.  ?  ^  of  2  s.  6  d.  ?  i  of  2  s.  4  d.  ?  ^  of  3  s. 

6d.?  fjyof  7s.  6d.?  Jof  l^d.?  ^of  2^d.r 


9.  At  1£,  5  s.  8|-d.  per 
yard,  what  will  6  yards  of 
cloth  cost  ? 


10.  If  6  yards  of  cloth  cost 
7£.  14  s.  4Jd.,  what  is  the 
price  per  yard  ? 


Here,  as  the  numbers  are  large,  it  will  be  most  convenient 
to  write  them  down  before  multiplying  and  dividing. 

OPERATION. 

£.    s.    d.  qr. 

6)7  14  4  2   cost  of  6  yards, 
3  price  of  1  yard. 


OPERATION. 

£.     s.  d.  qr. 

1     5  8  S  price  of  1  yard. 
6  number  of  yards. 

Ans.  7  14  4  2  cost  of  6  yards. 
6  times  3  qrs.  are  18  qrs.  = 
4  d.  and  2  qrs.  over ;  we  set 
down  the  2  qrs. ;  then,  6  times 
8d.  are  48  d.,  and  4  to  carry 
makes  52  d.  :zz  4  s.  and  4  d. 
over,  which  we  write  down ; 
again,  6  times  5  s.  are  30  s. 


1     5  8 

Proceeding  after  the  man- 
ner of  short  division,  6  is  con- 
tained in  7iS .  1  time,  and  1  £ . 
over ;  we  write  down  the 
quotient,  and  reduce  the  re- 
mainder (1^.)  to  shillings, 
(20  s.,)  which,  with  the  given 
shillings,  (14  s.,)  make  34  s.  j, 


94 


MULTIPLICATION    AND   DIVISION 


^41. 


and  4  to  carry  makes  34  s.  =: 
l£,  audl4s.  over;  6  times 
1^.  are  6jS.,  and  1  to  carry 
makes  7iB .,  which  we  write 
down  *,  and  it  is  plain,  that  the 
united  products  arising  from 
the  leveral  denominations  is 
the  real  product  arising  from 
the  whole  compound  number. 


11.  Multiply  3  jB.  4  s.  6  d. 
by  7. 

13.  What  will  be  the  cost 
of  5  pairs  of  shoes  at  10  s.  6  d. 
a  pair  ? 

15.  In  6  barrels  of  wheat, 
each  containing  2  bu.  3  pks. 
6  qts.,  how  many  bushels  r 

17.  How  many  yards  of 
cloth  will  be  required  for  9 
coats,  allowing  4  yds.  1  qr. 
3  na.  to  each  ? 

19.  In  7  bottles  of  wane, 
each  containing  2  qts.  1  pt.  3 
gills,  how  many  gallons  ? 

21.  What  will  be  the 
weight  of  8  silver  cups,  each 
weighing  5  oz.  12  pwt.  17 
grs.  ? 

23.  How  much  sugar  in  12 
hogsheads,  each  containing 
9cwt.  3qrs.  21  lb.  ? 

25.  In  15  loads  of  hay,  each 
weighing  1  T.  3  cwt.  2  qrs., 
how  many  tons  ? 


6  in  34  s.  goes  5  times,  and  4  s. 
over ;  4  s.  reduced  to  pence 
=  48  d.,  which,  with  the 
given  pence,  (4  d.,)  make  52 
d. ;  6  in  52  d.  goes  8  times,  and 
4  d.  over  ;  4  d.  =:  16  qrs., 
which,  with  the  given  qrs. 
(2)  =  18  qrs. ;  6  in  18 qrs.  goes 
3  times ;  and  it  is  plain,  that 
the  united  quotients  arising 
from  the  several  denomina- 
tions, is  the  real  quotient  aris-* 
ing  from  the  whole  compound 
number. 

12.  Divide  22 £.  lis.  6d. 
by  7. 

14.  At2ie.  12  s.  6  d.  for  5 
pairs  of  shoes,  what  is  that  a 
pair  ? 

16.  If  14  bu.  2  pks.  6  qts. 
of  wheat  be  equally  divided 
into  5  barrels,  how  many 
bushels  will  each  contain  ? 

18.  If  9  coats  contain  39 
yds.  3  qrs.  3na.,  what  does  1 
coat  contain  ? 

20.  If  5  gal.  1  gill  of  wine 
be  divided  equally  into  7  bot- 
tles, how  much  will  each  con- 
tain ? 

22.  if  8  silver  cups  weigh 
3  lb.  9  oz.  1  pwt.  16  grs.,  what 
is  the  weight  of  each  ? 

24.  If  119  c\vt.  1  qr.  of  su- 
gar be  divided  into  12  hogs- 
heads, how  much  will  each 
hogshead  contain  ? 

26.  If  15  teams  be  loaded 
withl7T.  12  cwt.  2  qrs.  of 
hay,  how  much  is  that  to  each 
team  ? 


ir4i. 


OF   COMPOUND   NUMBERS. 


95 


When  the  multiplier ^  or  divisor^  exceeds  12,  the  operations 
of  multiplying  and  dividing  are  not  so  easy,  unless  they  be 
composite  numbers ;  in  that  case,  we  may  make  use  of  the 
component  parts^  or  factors^  as  was  done  in  simple  numbers. 


Thus  15,  in  the  example 
above,  is  a  composite  number 
produced  by  the  multiplica- 
tion of  3  and  5,  (3x5  = 
15.)  We  may,  therefore, 
multiply  1  T.  3  cwt.  2  qrs.  by 
one  of  those  component  parts, 
or  factors,  and  that  product  by 
the  other,  which  will  give  the 
true  answer,  as  has  been  al- 
ready taught,  (IT  11.) 

OPERATION. 
T.  cwt.  qr. 
13     2 

3  one  of  the  factors. 

3     10     2 

5  the  other  factor. 


17     12     2   the  answer, 

27.  What  will  24  barrels 
of  flour  cost,  at  2ie .  12  s.  4  d. 
a  barrel  ? 

29.  What  will  112  lb.  of 
sugar  cost,  at  7^  d.  per  lb.  ? 

Note.  8,  7,  and  2,  are  fac- 
tors of  112. 

31.  How  much  brandy  in 
84  pipes,  each  containing  112 
gal.  2  qts.  1  pt.  3  g. .? 

33.  What  will  139  yards  of 
cloth  cost,  at  d  £,  6  s.  5 d. 
per  yard  ? 

139  is  not  a  composite  num- 
ber. We  may,  however,  de- 
compose this  number  thus, 
139  =  100+ 30 -f  9. 

We  may  now  multiply  the 


15  being  a  composite  num- 
ber, and  3  and  5  its  compo- 
nent parts,  or  factors,  we  may 
divide  17  T.  12  cwt.  2  qrs.  by 
one  of  these  component  parts, 
or  factors,  and  the  quotient 
thence  arising  by  the  other, 
which  will  give  the  true 
answer,  as  already  taught, 
(U20.) 

OPERATION. 

T.    cict.   qr. 

One  factor^        3)17    12    2 

The  other  factor,  6  )  5    17    2 

Ans.  13    2 


28.  Bought  24  barrels  of 
flour  for  62^.  16  s.  ;  how 
much  was  that  per  barrel  ? 

30.  If  1  cwt.  of  sugar  cost 
3  £,  7  s.  8  d.,  what  is  that  per 
lb,  ? 

32.  Bought  84  pipes  of 
brandy,  containing  9468  gal. 
1  qt.  1  pt.  ;  how  much  in  a 
pipe  ? 

34.  Bought  139  yards  of 
cloth  for  461^.  lis.  lid.; 
what  was  that  per  yard  ? 

When  the  divisor  is  such  a 
number  as  cannot  be  produced 
by  the  multiplication  of  small 
numbers,  the  better  way  is  to 
divide  after  the  manner  of 


96 


MULTIPLICATION    ANB   DIVISION 


IT  41, 


price  of  1  yard  liy  10,  which 
will  give  the  price  of  10  yards, 
and  this  product  again  by  10, 
which  will  give  the  price  of 
100  yards. 

We  may  then  multiply  tlae 
price  of  10  yards  by  3,  which 
will  give  the  price  of  30  yards, 
and  the  price  of  1  yard  by  9, 
which  will  give  the  price  of 
9  yards,  and  these  three  pro- 
ducts, added  together,  will  evi- 
dently give  the  price  of  139 
yards ;  thus  : 

£.      s.      d. 

3      6      5  price  of  1  yd, 
10 

price  of  10  yds, 

8  price  of  100  yds. 
price  of  30  yds, 
price  of     9  yds, 

461  11  11  price  of  139  yds. 
Note,  In  multiplying  the 
price  of  10  yards  (33£.  4  s. 
2d.)  by  3,  to  get  the  price  of 
30  yards,  and  in  multiplying 
the  price  of  1  yard  (^£,  6  s. 
5  d.)  by  9,  to  get  the  price  of 
9  yards,  the  multipliers,  3  and 
9,  need  not  be  written  down. 


33 

4 

2 

10 

332 

1 

8 

99 

12 

6 

29 

17 

9 

but  may 
mind. 


be    carried   in   the 


long  division,  setting  down 
the  work  of  dividing  and  re- 
ducing in  manner  as  fol- 
lows : 

139)46i     11     U{Z£, 
417 

44 
20 

891  (6  s. 
834 

12 

695  (  5  d. 
695 

The  divisor,  139,  is  con- 
tained in  461  £,  3  times, 
(3i£.,)  and  a  remainder  of 
44£.,  which  must  now  be 
reduced  to  shillings,  multi- 
plying it  by  20,  and  bringing 
in  the  given  shillings,  (lis.,) 
making  891  s.,  in  which  the 
divisor  is  contained  6  times, 
(6  s.,)  and  a  remainder  of 
57  s.,  which  must  be  reduced 
to  pence,  multiplying  it  by  12, 
and  bringing  in  the  given 
pence,  (11  d.,)  together  mak- 
ing 695  d.,  in  which  the  di- 
visor is  contained  5  times, 
(5  d.,)  and  no  remainder. 

The  several  quotients,  S£ , 
6  s.,  5  d.,  evidently  make  the 
answer. 


V4l. 


OF    COMPOUND   IfUMBCRS. 


97 


The  processes  in  the  foregoing  examples  may  now  be  pre- 
sented in  the  form  of  a 


Rule  for  the  Multiplication  oj 
Compound  Numbers. 
I.  When  the  multiplier  does 
not  exceed  12,  multiply  sue 
cessively  the  numbers  of  each 
denomination,  beginning  with 
the  least,  as  in  multiplication 
of  simple  numbers,  and  carry 
as  in  addition  of  compound 
numbers,  setting  down  the 
whole  product  of  the  highest 
denomination. 


II.  If  the  multiplier  exceed 
12,  and  be  a  composite  num- 
ber, we  may  multiply  first  by 
one  of  the  component  parts, 
Ihat  product  by  another,  and 
80  on^  if  the  component  parts 
be  more  than  two ;  the  last 
product  will  be  the  product  re- 
quired. 

III.  When  the  multiplier 
exceeds  12,  and  is  not  a  com- 
posite, multiply  first  by  10, 
and  this  product  by  10,  whi^h 
will  give  the  product  for  100; 
and  if  the  hundreds  in  the?mul- 
tiplier  be  more  than  one,  ravi' 
tiply  the  product  of  100  by  the 
number  of  hundreds ;  for  the 
tens^  multiply  the  product  of 
10  by  the  number  of  tens ;  for 
the  units^  multiply  the  mulii- 
vlicand;  and  these  several  pro- 
ducts will  be  the  product  re- 
quired. 

1 


Rule  for  the  Division  of  Cam* 
pound  Numbers. 

I.  When  the  divisor  does 
not  exceed  12,  in  the  manner 
of  short  division,  find  how 
many  times  it  is  contained  in 
the  highest  denomination,  un- 
der which  write  the  quotient, 
and,  if  there  be  a  remainder, 
reduce  it  to  the  next  less  de- 
nomination, adding  thereto  the 
number  given,  if  any,  of  that 
denomination,  and  divide  as 
before  ;  so  continue  to  do 
through  all  the  denominations, 
and  the  several  quotientsi^  will 
be  the  answer. 

II.  If  the  divisor  exceed  12, 
and  be  a  composite^  we  may  di- 
vide first  by  one  of  the  com- 
ponent parts,  that  quotient  by 
another,  and  so  on,  if  the  com- 
ponent parts  be  more  than 
two  ;  the  last  quotient  will  be 
the  quotient  required. 

III.  When  the  divisor  ex- 
ceeds 12,  and  is  not  a  com- 
posite pumber,  divide  after  the 
manBcr  of  long  division,  set- 
ting down  the  work  of  di- 
Tiding  and  reducing. 


MULTIPUCATION    ANP    DITISION,    &C,  IT 


EXAMPLES  FOR   PRACTICE. 


1.  What  will  359  yards  of 
cloth  cost,  at  4  s.  7^  d.  per 
yard  ? 

3.  In  241  barrels  of  flour, 
6ach  containing  1  (iwt.  3  qr. 
9  lb. ;  how  many  cvvt.  ? 

6.  How  many  bushels  of 
wheat  in  135  bags,  each  con- 
taining 2  bu.  3  pks.  ? 

3X9X5  =  135. 

7.  What  will  35  cwt.  of  to- 
bacco cost,  at  3  s.  104-  tl.  per 
lb.  ? 

9.  If  14  men  build  12  rods 
6  feet  of  wall  in  one  day,  how 
many  rods  will  they  build  in 
7^  days  ?     . 


2.  Bought  359  yards  of  cloth 
for  83  jg .  0  s.  4i  d. ;  what  was 
that  a  yard  ? 

4.  If  441  cwt.  13  lb.  of  flour 
be  contained  in  241  barrels, 
how  much  in  a  barrel  ? 

6.  If371bu.  Ipk.  of  wheat 
be  divided  equally  into  135 
bags,  how  much  will  each  bag 
contain  ? 

5.  At  759  i£.  10  3.  for  35 
cwt.  of  tobacco,  what  is  that 
per  lb. ? 

10.  If  14  men  build  92  rods 
12  feet  of  stone  wall  in  7^ 
days,  how  much  is  that  per 
day? 


IT  42.    1.  At  10  s.  per  yard,  what  will  17849  yards  of 
cloth  cost  ? 

Notc^  Operations  in  multiplication  of  pounds,  shillings, 
pence,  or  of  any  compound  numbers,  may  be  facilitated  by 
taking  aliquot  parts  of  a  higher  denomination^  as  already  ex- 
plained in  ^^Practice^^  of  Federal  Money,  1129,  ex.10. 
Thus,  in  this  last  example,  the  price  10  s.  =:^  of  a  pound  ; 
therefore,  ^  of  the  number  of  yards  will  be  the  cost  in 
pounds.     -i-^|49.  — 8924  £,  10s.  Ans. 

2.  What  cost  31648  yards  of  cloth,  at  10  s.  or  JiS.  per 

yard  ?     at  5  s.  ^  ^£ .  per  yard  ?     •  at  4  s.  =  -^^ , 

per  yard  f    > at  3  &.  4  d.  :==  ^£ .  per  yard  ?    at  2  s. 

r=  ^V  ^  •  per  yard  ?  Am.  to  last^  34g4  iS .  16  s. 

3.  What  cost  7430  pounds  of  sugar,  at  6  d.  z=i  J  s.  per  lb  ? 

at  4  d.  =  ^  s.  per  lU?     at  3d.  :=  i  s.  per 

lb.?      at  2  d.=:^  s.  per  «>.? at    l^'d.=:^s. 

per  lb.  ?  J|k 

Ans,  to  the  last^  i4^ff|.  =  928  s.  9  d.  =  46  iS .  8  s.  9  d. 

4.  At  $18'75  per  cwt.,  what  will  2qrs.  =^cwt.  cost? 

what  will  1  qr.  =  ^  cwt.  cost  ? .  what  will  16  lb, 

=  f  cwt.  cost  ?  what  will  14  lbs.  ==  ^  cwt.  cost  ?    

what  will  8  lbs.  =  -ifjf  cwt.  cost  ?     Ans.  to  the  last ^$^^^9- 


t  42.         SUPPLEMENT   TO   COMPOUND   NUMBERS.  9& 

6.  What  cost  340  yards  of  cloth,  at  12  s.  6  d.  per  yard  ? 
12s.6d.  =  10s.  {z=i£.)  and2s.6d.  (zz^ig.);  there- 
fore, 

i)i)S40 

170  iB .  =  cost  at  10  s.  per  yard. 

42  iS .  10  s.  =  at  2  s.  6  d.    per  yard. 

Ans,  212  ig   10  s.  =  at  12  s.  6  d.  peryard. 

Or, 
10  s.  z=  Jig.)  340 

S  8.  6  d.  =  i  of  10  s.)  170  ig .         at  10  s.  per  yard. 

42  jB .  10  s.  at  2  s.  6  d.  per  yard. 

Ans.  212  ie .  10  s.  at  12  s.  6  d.  per  yard. 


SUPPZiZSMENT    TO    THS    ARXTHHHSTXC  OF 
COXMCPOUND    NUMBERS. 

QUESTIONS. 

1.  What  distinction  do  you  make  between  simple  and 
compound  numbers  ?  (^  26.)  2.  What  is  the  rule  for  addi- 
tion of  compound  numbers  ?     3.  for  subtraction  of,  &c.  ? 

4.  There  are  three  conditions  in  the  rule  given  for  multi- 
plication of  compound  numbers;  what  are  they,  and  the 
methods  of  procedure  under  each?  5.  The  same  questions 
in  respect  to  the  division  of  compound  numbers  ?  6.  When 
the  multiplier  or  divisor  is  encumbered  with  a  fraction,  how 
do  you  proceed  ?  7.  How  is  the  distance  of  time  from  one 
iate  to  another  found  ?  8.  How  many  degrees  does  the 
earth  revolve  from  west  to  east  in  1  hour?  9.  In  what 
time  dees  it  revolve  1°?  Where  is  the  time  or  hour  of  the 
day  earlier — at  the  place  most  easterly  or  most  westerly  ? 
10.  The  difference  in  longitude  between  two  places  being 
known,  how  is  the  difference  in  Jime  calculated  ?  11.  How 
may  operations,  in  the  multiplication  of  compound  num- 
bers, be  facilitated  ?     12.  What  are  some  of  the  aliquot  parts 

ofil£.?    ofls.?    oflcwt?     13.  Whatisthi« 

manner  of  operating  usually  called  ? 


100  SUPPLSMBNT   TO   COMPOUND   NUMBERS.         TT  40. 


£X£RCIS£S. 

1.  A  gentleman  is  possessed  of  1 J  dozen  of  silver  spoons, 
each  weighing  3  oz.  5  put. ;  2  doz.  of  tea  spoons,  each  weigh- 
ing 15  pwt.  14  gr. ;  3  silver  cans,  each  9  oz.  7  pwt. ;  2  silver 
tankards,  each  21  oz.  15  pwt. ;  and  6  silver  porringers,  each 
11  oz.  18  pwt. ;  w^hat  is  tlie  weight  of  the  whole  ? 

Ans.  18  lb.  4  oz.  3  pwt- 

Note,  Let  the  pupil  be  required  to  reverse  and  prove  the 
following  examples : 

2.  An  English  guinea  sh-^uld  weigh  5  pwt.  6  gr. ;  a  piece 
of  gold  weighs  3  pwt.  IT  gr. ;  how  much  is  that  short  of  the 
weight  of  a  guinea  ? 

3.  What  is  the  weight  of  6vchests  of  tea,  each  weighing 
3  cwt.  2  qrs.  9  lb.  ? 

4.  In  35  pieces  of  cloth,  each  measuring  27  yards,  how 
many  j^ards  ? 

5.  How  much  brandy  in  9  casks,  each  containing  45  gal. 
3  qts.  1  pt.  ? 

6.  If  31  cwt.  2  qrs.  20  lb.  of  sugar  be  distributed  equally 
into  4  casks,  how  much  will  each  contain  ? 

7.  At  4^  d.  per  lb.,  what  costs  1  cwt  of  rice  ?  2  cwt  ^ 

3  cwt  ? 

Note.  The  pupil  will  recollect,  that  8,  7  and  2  are  fac- 
tors of  112,  and  may  be  used  in  place  of  that  number. 

8.  If  800  cwt  of  cocoa  cost  18  iS .  13  s.  4  d.,  what  is  that 
per  cwt.  ?     what  is  it  per  lb.  ? 

9.  What  will  9^  cwt.  of  copper  cost  at  5  s.  9  d.  per  lb.  ? 

10.  If  6^ cwt  of  chocolate  cost  72  i£.  16  s.,  what  is  that 
per  lb.  ? 

11.  What  cost  456  bushels  of  potatoes,  at  2  s.  6d.  per 
bushel  ? 

Note.     2  s.  6  d.  is  ^  of  1  £ .     (See  IT 42.) 

12.  What  cost  86  yards  of  broadcloth,  at  15s.  per  yard? 
Note.     Consult  ^  42,  ex.  5. 

13.  What  cost  7846  pounds  of  tea,  at  7  s.  6  d.  per  lb.  ? 
at  14  s.  per  lb.  ?     at  13  s.  4  d.  ? 

14.  At  $  94'25  per  cwt,  what  will  be  the  cost  of  2  qrs, 

of  tea?     of  3  qrs.  ?    of  14  lbs.  ?     of  21  lbs,? 

of  16  lbs.  ?     of  24  lbs.  ? 

NoU.     Consult  IT  42,  ex.  4  and  6. 


T42,43.  FRACTIONS.  10! 

15.  What  will  be  the  cost  of  2  pks.  aud  4  qts.  of  wheat, 
at  $  1*50  per  bushel  ? 

16.  Supposing  a  meteor  to  appear  so  high  in  the  heavens 
as  to  be  visible  at  Boston,  71°  3',  at  the  city  of  Washington, 
IT  43',  and  at  the  Sandwich  Islands,  155°  W.  longitude, 
and  that  its  appearance  at  the  city  of  Washington  be  at  7 
minutes  past  9  o'clock  in  the  evening;  what  will  be  the 
hour  and  minute  of  its  appearance  at  Boston  and  at  the 
Sandwich  Islands  ? 


FRACTION'S. 

^  43.  We  have  seen,  (IT  17,)  that  numbers  expressing 
whole  things  are  called  integers,  or  whole  numbers ;  but  that, 
in  division,  it  is  often  necessary  to  divide  or  break  a  whole 
thing  into  parts,  and  that  these  parts  are  called  f-actions^  or 
broken  numbers. 

It  will  be  recollected,  (IT  14,  ex.  11,)  that  when  a  thing 
or  unit  is  divided  into  3  parts,  the  parts  or  fractions  are  call- 
ed thirds  ;  when  into  four  parts,  fourths  ;  when  into  six  parts, 
nxths  ;  that  is,  the  fraction  takes  its  name  or  denwnination  from 
the  number  of  parts,  into  which  the  unit  is  divided.  Thus, 
if  the  unit  be  divided  into  16  parts,  the  parts  are  called  six- 
teenths, and  5  of  these  parts  ^^  ould  be  5  sixteenths,  expressed 
thus,  -3^.  The  number  below  the  short  line,  (16,)  as  before 
taught,  (IT  17,)  is  called  the  denominator,  because  it  gives 
the  name  or  denomination  to  the  parts ;  the  number  above 
tlie  line  is  called  the  mimerator,  because  it  numbers  the  parts. 

The  denominator  shows  how  many  parts  it  takes  to  make 
^unit  or  whole  thing;  the  numerator  shows  how  many  of 
these  parts  are  expressed  by  the  fraction, 

1.  If  an  orange  be  cut  into  5  equal  parts,  by  what  frac- 
tion is  1  part  expressed  ?     2  parts  }     3  parts  ? 

4  parts  ?     5  parts  ?   how  many  parts  make  unity 

or  a  whole  orange  ? 

2.  If  a  pie  be  cut  into  8  equal  pieces,  and  2  of  these 
pieces  be  given  to  Harry,  what  will  be  his  fraction  of  the 
pie  ?  if  5  pieces  be  given  to  John,  what  will  be  his  fraction  } 
what  fraction  or  part  of  the  pie  will  be  left  ? 

It  is  important  to  bear  in  mind,  that  fractions  arise  from 
division^  (IF  17,)  and  that  the  nviaerator  may  be  considered  a 
I  * 


J  expresses  the  quotient,  of  which  J 


102  FRACTIONS.  It  4 J. 

dividend^  and  the  denominator  a  divisor,  and  the  tjo/ttc  of  the 
fraction  is  the  quotient ;  thus,  5-  is  the  quotient  of  1  (the 
numerator)  divided  by  2,  (the  denominator ;)  J  is  the  quo- 
tient arising  from  1  divided  by  4,  and  f  is  3  times  as  much, 
that  is,  3  divided  by  4 ;  thus,  one  fourth  part  of  3  is  the 
same  as  3  fourths  of  1 . 

Hence,  in  all  cases,  a  fraction  is  always  expressed  by  the 
sign  oj  divhion, 

3  is  the  dividend^  or  numerator  . 

4  is  the  divisor f  or  denominator. 

g.  If  4  oranges  be  equally  divided  among  6  boys,  what 
part  of  an  orang**.  is  each  boy's  share  ? 

A  sixth  part  of  1  orange  is  ^,  and  a  sixth  part  of  4  oranges 
is  4  such  pieces,  =  |.  Ans.  |  of  an  orange. 

4.  If  3  apples  be  equally  divided  among  5  boys,  what  part 
of  an  apple  is  each  boy's  share  ?  if  4  apples,  what  ?  if  2 
apples,  what?     if 5  apples,  what? 

5.  What  is  the  quotient  of  1  divided  by  3?    of  2  by  3? 

of  1  by  4?    of2by4?    of3by4?    of5 

by7?     ofebyS?    of  4  by  5  ?    of2byl4? 

6.  What  part  of  an  orange  is  a  third  part  of  2  oranges  ? 

one  fourth  of  2  oranges  ?     J  of  3  oranges  ?     

I  of  3  oranges? iof4?     i  of 2  ?     |of6? 

I  of  3  ?     — —  :J-  of  2  ? 

^4  Proper  Fraction,  Since  the  denominator  shows  the  num- 
ber of  parts  necessary  to  make  a  whole  thing,  or  1,  it  is  plain, 
tliat,  when  the  numerator  is  less  than  the  denominator,  the 
fraction  is  less  than  a  unit,  or  whole  thing  ;  it  is  then  called  a 
proper  fraction.     Thus,  ^,  f ,  ^c.  are  proper  fractions. 

An  Improper  Fraction,  When  the  numerator  equals  or  ez- 
ceeds  the  denominator,  the  fraction  equals  or  exceeds  unity,  or 
1,  and  is  then  called  an  improper  fraction.  Thus,  f,  f,  f,  -y^, 
are  improper  fractions. 

A  Mixed  Number,  as  already  shown,  is  one  composed  of  a 
whole  number  and  a  fraction.  Thus,  14^,  13  J,  &:c.  are  mix* 
ed  numbers. 

7.  A  father  bought  4  oranges,  and  cut  each  orange  into  d 
equal  parts ;  he  gave  to  Samuel  3  pieces,  to  James  5  pieces^ 
to  Mary  7  pieces,  and  to  Nancy  9  pieces ;  what  was  each 
one's  fraction  ? 

Was  Jameses  fraction  proper,  or  improper  1    Why  ? 

Was  Nancy's  fraction  proper,  or  improper  ?    Why  ? 


IT  44.  FRACTIONS-  103 

To  change  an  improper  fraction  to  a  whole  or  mixed  number, 

IT  44.  It  is  evident,  that  every  improper  fraction  must 
contain  one  or  more  whole  ones,  or  integers. 

1.  How  many  whole  apples  are  there  in  4  halves  (J)  of 

8in  apple  ?     in  f  ?    in  f  ?    in  -^^  ?     in 

^?     in-^?     inJ^I^?     in^? 

2.  How  many  yards  in  f  of  a  yard  ?     in  §  of  a  yard  ? 

inf?     — ^ — inf?   in-^-?     in-y^?   in 

A^P     inJgL?     in^<i?     in  4/? 

3.  How  many  bushels  in  8  pecks  ?  that  is,  in  |  of  a  bushel  ? 

inJj^-? inJ^?   in-\^?   in ^4^?   ^in 

x^  ?    in  -^  ? 

This  finding  how  many  integers,  or  whole  things,  are  con- 
tained in  any  improper  fraction,  is  called  reducing  an  impro^ 
per  fraction  to  a  whole  or  mixed  number, 

4.  If  I  give  27  children  ^  of  an  orange  each,  how  many 
oranges  will  it  take  ?  It  will  take  ^ ;  and  it  is  evident,  that 
OPERATION  dividing  the  numerator,  27,  (im  the  Rum- 

^ \  27  ber  of  parts  contained  in  the  fraction, )  by 

1 the  denominator,  4,  (=:  the  number  of 

Ans,  6 J-  oranges,         parts  in  1  orange,)  will  give  the  number 
of  whole  oranges. 
Hence,  To  reduce  an  improper  fraction  to  a  whole  or  mixed 
number^ — Rule  :  Divide  the  numerator  by  the  denominator ; 
the  quotient  will  be  the  whole  or  mixed  number. 

EXAMPluES   FOR   PRACTICE. 

*  o.  A  mail,  spending  -J  of  a  dollar  a  day,  in  83  days  would' 
spend  ^  of  a  dollar  ;  how  many  dollars  would  that  be  ? 

Ans,  $13f 
6.  In  -^li^  of  an  hour,  how  many  whole  hours  ? 
The  60th  part  of  an  hour  is  1  minute :  therefore  the  ques- 
tion is  evidently  the  same  as  if  it  had  been,  In  1417  minutes, 
how  many  hours  ?  Ans,  23 IJ-  hours, 

vl  7.  In  -^^l?-  of  a  shilling,  how  many  units  or  shillings  ? 

Ans,  730^^  shillings. 

8.  Reduce  -^P-  to  a  whole  or  mixed  number. 

9.  Reduce  f§,  ^e,  fj^,  ^Jff,  ^^,  to  whole  or  Bail- 
ed numbers. 


104  FRACTIONS.  IT  43, 

To  reduce  a  whole  or  mixed  number  to  an  improper  fraction. 

IT  45.  We  have  seen,  that  an  improper  fraction  may  be 
changed  to  a  whole  or  mixed  number ;  and  it  is  evident, 
that,  by  reversing  the  operation,  a  whole  or  mixed  number 
may  be  changed  to  the  form  of  an  improper  fraction. 

1.  In  2  i^jAo/e  apples,  how  many /laZres  of  an  apple?  Ans.  4 
halves ;  that  is,  f .  In  3  apples,  how  many  halves  ?  in  4 
apples  ?  in  6  apples  ?  in  10  apples  ?  in  24  ?  in  60  ?  in 
170  ?     in  492  ? 

2.  Reduce  2  yards  to  thirds,  Ans.  f .     Reduce  2f  yards  to 

thirds.    Ans.  f .     Reduce  3  yards  to  thirds.     3^  yards. 

3f  yards.     5  yards.     5f  yards.     6f 

yards. 

3.  Reduce  2  bushels  to  fourths, 2f  bu. 6  bushels. 

6^  bushels.     7f  bushels.     25f  bushels. 

4.  In  16-fj  dollars,  how  many  y'^-  ^^  ^  dollar? 

^f  make  1  dollar:  if,  therefore,  we  multiply  16  by  12,  that 
is,  multiply  the  ichole  number  by  the  denominator^  the  product 
will  be  the  number  of  12ths  in  16  dollars  :  16  X  12  =  192, 
and  this,  increased  by  the  numeiator  of  the  fraction,  (5,)  evi- 
dently gives  the  whole  number  of  12ths ;  that  is,  ^^  of  a 
dollar.  Answer, 

OPERATION. 
16^V  dollars, 

12  ": 


192  z=  12ths  in  16  dollars,  or  the  whole  number- 
*  5  z=  12ths  contained  in  ih^  fractiori, 

197  z-  -y^T-j  the  answeu 
Hence,  To  reduce  a  mixed  number  to  an  improper  fraction^ — 
Rule  :  Multiply  the  whole  number  by  the  denominator  of 
the  fraction,  to  the  product  add  the  numerator,  and  write 
the  result  over  the  denominator. 

exampi.es  for  practice. 

lb.  What  is  the  improper  fraction  equivalent  to  23fJ  hours  ? 

Ans,  ^%^^  of  an  hour. 
'  6.  Reduce  730-^  shillings  to  I2ths. 
As  -j^  of  a  shilling  is  equal  to  1  penny,  the  question  is  evi- 
dently the  same  as.  In  730  s.  3  d.,  how  many  pence  ? 

Ans.  -^p-  of  a  shilling ;  that  is,  8763  penc«. 


IT  45, 46.  FRACTIONS.  05 

7.  Reduce  1  j§,  17f  |,  8^^,  4^^,^^,  and  1^  to  improper 
fractions. 

8.  In  156^J  days,  how  many  24ths  of  a  day? 

Ans.  ij^p  =  3761  hours. 

9.  In  342 J  gallons,  how  many  4ths  of  a  gallon  ? 

Ans.   'y  >-  of  a  gallon  =  1371  quarts. 

To  reduce  a  fraction  to  its  lowest  or  most  simple  terms, 

^  46.  The  numerator  and  the  denominator,  taken  to- 
gether, are  called  the  terms  of  the  fraction. 

If  ^  of  an  apple  be  divided  into  2  equal  parts,  it  becomes  f . 
The  effect  on  the  fraction  is  evidently  the  same  as  if  we  had 
multiplied  both  of  its  terms  by  2.  In  either  case,  ike  parts 
are  made  2  times  as  many  as  they  were  before  ;  hut  they  are  only 
HALF  AS  LARGE ;  for  it  will  take  2  times  as  many  fourths  to 
make  a  whole  one  as  it  will  take  halves ;  and  hence  it  is 
that  f  is  the  same  in  value  or  quantity  as  ^. 

f  is  2  parts ;  and  if  each  of  these  parts  be  again  divided 
into  2  equal  parts,  that  is,  if  both  terms  of  the  fraction  be 
multiplied  by  2,  it  becomes  -|.  Hence,  ^  =  J  z=  |,  f  5  "^e 
reverse  of  this  is  evidently  true,  that  -|  izz  f  nr  J.  ' ^^ 

It  follows  therefore,  by  multiplying  or  dividing  both  terms  of 
the  fraction  by  the  same  number^  we  chc^'ge  its  terms  without 
altering  its  value. 

Thus,  if  we  reverse  the  above  operauon,  and  divide  both 
terms  of  the  fraction  |  by  2,  we  obtain  its  equal,  f  ;  dividing 
again  by  2,  we  obtain  ^,  which  is  the  most  simple  form  of  th  t 
fraction,  because  the  terms  are  the  least  possible  by  which 
the  fraction  can  be  expressed. 

The  process  of  changing  |  into  its  equal  J  is  called  re- 
ducing  the  fraction  to  its  lowest  terms.  It  consists  in  dividing 
both  terms  of  the  fraction  by  any  ''umber  which  will  divide  them 
both  without  a  remainder ^  and  the  quotient  thence  arising  in  the 
same  manner ^  and  so  on^  till  it  apptars  that  no  number  greater 
tfian  1  will  again  divide  them, 

A  number,  which  will  divide  two  or  more  numbers  with- 
out a  remainder,  is  called  a  common  divisor,  01  common  meor 
Buxe  of  those  numbers.  The  greatest  numbei  that  will  do 
this  is  called  the  greatest  common  divisor. 


106  FRACTIONS.  fr  46,  41 

1.  What  part  of  an  sere  are  128  rods  ? 

One  rod  is  j^  of  an  acre,  and  128  rods  are  -Jff  of  an 
acre.  Let  us  reduce  this  fraction  to  its  loivest  terms.  We 
find,  hy  trial,  that  4  will  exactly  measure  both  128  and  160, 
and,  dividing,  we  change  the  fraction  to  its  equal  Jg.  Again, 
we  find  that  8  is  a  divisor  common  to  both  terms,  and,  di- 
viding, we  reduce  the  fraction  to  its  equal  |^,  which  is  uoxt 
in  its  lowest  terms,  for  no  greater  number  than  1  will  again 
measure  them.     The  operation  may  be  presented  thus  : 

^>.128   ^2      4     .  . 

2.  Reduce  f^g,  j^^,  ||-g-,  and  ^||  to  their  lowest  terms. 

^^'  h  h  h  ^^^  i- 

Note,  If  any  number  ends  with  a  cipher,  it  is  evidently 
divisible  by  10.  If  the  two  right  hand  figures  are  divisible 
by  4,  the  whole  number  is  also.  If  it  ends  with  an  even 
number^it  is  divisible  by  2 ;  if  with  a  5  or  0,  it  is  divisible 

^^/Ihis.; 

"i^K       uce  ^J|,  ^,  ifl,  and  f ^  to  tlieir  lowest  terms. 

1^7.  Any  fraction  may  evidently  be  reduced  to  its  lowest 
terms  by  a  single  division,  if  we  use  the  greatest  common 
divisor  of  the  two  terms.  The  greatest  common  measure  of 
any  two  numbers  may  be  found  by  a  sort  of  trial  easily  made* 
Let  the  numbers  be  tSic  two  terms  of  the  fraction  -fff .  The 
common  divisor  cannot  esiceed  the  less  number,  for  it  must 
measure  it.  We  ^vill  try,  therefore,  if  the  less  number,  128, 
which  measures  itself,  will  also  divide  or  measure  160. 

128^160('l  ^^^  ^^  ^^^  ^^^^  ^  time,  and  32  re- 

■loa  mam;  128,  therefore,  is  not  a  divisor  of 

160.     We  will  now  try  whether  this  re- 

32)  128(4         mainder  be  not  the  divisor  sought ;  for  if 
128  32  be  a  divisor  of  128,  the  former  divi- 

sor,  it  must  also  be  a  divisor  of  160, 

which  consists  of  128  -[-  32.  32  in  128  goes  4  times,  with- 
out any  remainder.  Consequently,  32  is  a  divisor  of  128  and 
160.  And  it  is  evidently  the  greatest  common  divisor  of 
these  numbers ;  for  it  must  be  contained  at  least  once  more  in 
160  than  in  128,  and  no  number  greater  than  their  difference, 
that  is,  greater  than  32,  can  do  it. 


IT  47, 48.  FRACTIONS  107 

Hence  the  nth  for  finding  the  greatest  common  dioisor  of 
iwo  numbers : — Divide  the  greater  number  by  the  less,  and 
that  divisor  by  the  remainder,  and  so  on,  always  dividing 
the  last  divisor  by  the  last  remainder,  till  nothing  remain. 
The  last  divisor  will  be  the  greatest  common  divisor  required. 

Note*  It  is  evident,  that,  when  w^e  would  find  the  greatest 
common  divisor  of  inore  than  two  numbers,  we  may  first  find 
the  greatest  common  divisor  of  two  numbers,  and  then  of 
that  common  divisor  and  one  of  the  other  numbers,  and  so 
on  to  the  last  number.  Then  will  the  greatest  common 
divisor  last  found  be  the  answer. 

4.  Find  the  greatest  common  divisor  of  the  terms  of  the 
fraction  f  ^,  and,  by  it,  reduce  the  fraction  to  its  lowest  terms. 

OPERATION. 

21)35(1 

21 

14)21(1 
14 

Greatest  divis.    7)  14(2.  Then,  7)—=:^  An$, 

14  ^35       5 

6.  Reduce  ^^^^  to  its  lowest  terms.  Am.  ^. 

Ao/e.  Let  these  examples  be  wrought  by  both  methods ; 
by  several  divisors,  and  also  by  finding  the  greatest  common 
divisor.  ^?     • 

f  6.  Reduce  -ff^r  to  its  lowest  terms.  Ans,  -J, 

f  7.  Reduce  J^f  to  its  lowest  terms.  ^  Ans.  f, 

8.  Reduce  -^t%-  to  its  lowest  terms.  Ans,  ^^ 

9.  Reduce  J-f  f |  to  its  lowest  terms.  Ans,  ^, 

To  divide  a  fraction  by  a  whole  number, 
IT  48.    1.  If  2  yards  of  cloth  cost  f  of  a  dollar,  what  does 
1  yard  cost  ?  how  much  is  f  divided  by  2  ? 

2.  If  a  cow  consume  f  of  a  bushel  of  meal  in  3  days,  how 
much  is  that  per  day  ?  J  -i-  3  z=  how  much  ? 

3.  If  a  boy  divide  |  of  an  orange  among  2  boys,  how  much 
will  he  give  each  one  ?  |  -f-  2  z=  how  much  > 

4.  A  boy  bought  5  cakes  for  ^J  of  a  dollar ;  what  did  I 
cake  cost  ?  -ff  -r-  5  =  how  much  ? 


108  TRACTIONS.  48    t 

5.  If  2  bushels  of  apples  cost  f  of  a  dollar,  what  is  that 
per  bushel  ? 

1  bushel  is  the  half  of  2  bushels  j  the  half  of  f  is  ^. 

Ans.  ^  dollar. 

6.  If  3  horses  consume  ^f  of  a  ton  of  hay  in  a  month, 
what  will  1  horse  consume  in  the  same  time  ? 

^f  are  12  parts ;  if  3  horses  consume  12  such  parts  in  a 
month,  as  many  times  as  3  are  contained  in  12,  so  many 
parts  1  horse  will  consume.  A7is.  ^  of  a  ton. 

7.  If  f  I  of  a  barrel  of  flour  be  divided  equally  among  5 
families,  how  much  will  each  family  receive  ? 

f|-  is  25  parts ;  5  into  25  goes  5  times.     Ans.  -^^  of  a  barreL 

The  process  in  the  foregoing  examples  is  evidently  di- 
viding a  fraction  by  a  whole  number ;  and  consists,  as  may 
be  seen,  in  dividing  the  numerator^  (when  it  can  be  done 
without  a  remainder,)  and  under  the  quotient  writing  the 
denominator.  But  it  not  unfrequently  happens,  that  the  nu- 
merator will  not  contain  the  whole  number  without  a  re- 
mainder. 

8.  A  man  divided  J  of  a  dollar  equally  among  2  persons  ; 
what  part  of  a  dollar  did  he  give  to  each  ? 

^  of  a  dollar  divided  into  2  equal  parts  will  be  4ths. 

Ans.  He  gave  J  of  a  dollar  to  each. 

9.  A  mother  divided  -J-  a  pie  among  4  children  ;  what  part 
of  the  pie  did  she  give  to  each  ?  ^  --  4  :::=  how  much  ? 

10.  A  boy  divided  -J  of  an  orange  equally  among  3  of  his 
companions;  what  was  each  one's  share?  -J- -f- 3  =  how 
much  ? 

11.  A  man  divided  f  of  an  apple  equally  between  2  chil- 
dren ;  what  part  did  he  give  to  each  ?  f  divided  by  2  =: 
what  part  of  a  whole  one  ? 

J  is  3  parts  :  if  each  of  these  parts  be  divided  into  2  equal 
parts,  they  will  make  6  parts.  He  may  now  give  3  parts  to 
one,  and  3  to  the  other  :  but  4ths  divided  inio  2  equal  parts, 
become  8tlis.  The  parts  are  now  twice  so  many^  but  they 
are  only  half  so  large  ;  consequently,  f  is  only  half  so  much 
as  f .  Ansi.  f  of  an  apple. 

In  these  last  examples,  the  fraction  has  been  divided  by 
multiplying  the  denominator^  without  char.ging  the  numerator. 
The  reason  is  obvious ;  for,  by  multiplying  the  denominator 
bj  any  number,  the  parts  are  made  so  many  times  smaller^ 
amce  it  will  take  so  many  more  of  them  to  make  a  whole 


IT  49,  FRACTIONS.  109 

pne ;  and  if  no  more  of  these  smaller  parts  be  taken  than 
were  before  taken  of  the  larger,  that  is,  if  the  numerator  be 
not  changed,  the  value  of  the  fraction  is  evidently  made  so 
many  times  less. 

IT  49.  Hence,  vi^e  have  two  ways  to  divide  a  fraction  by 
a  whole  number : — 

I.  Divide  the  numerator  by  the  whole  number,  (if  it  will 
contain  it  without  a  remainder,)  and  under  the  quotient  write 
the  denominator. — Otherwise, 

II.  Multiply  the  denominator  by  the  whole  number,  and 
over  the  product  write  the  numerator. 

EXAMPLES   FOR   PRACTICE. 

1.  If  7  pounds  of  coiFee  cost  f^  of  a  dollar,  what  is  that 
per  pound  ?  f-J-  -f-  7  iz:  how  much  ?  Ans,  ^  of  a  dollar. 

2.  If  ^  of  an  acre  produce  24  bushels,  what  part  of  an 
acre  will  produce  1  bushel  ?  ^§  ~  24  =:  how  much  ? 

3.  If  12  skeins  of  silk  cost  ^J  of  a  dollar,  what  is  that  a 
«kein  ?  ^  -f.  12  =  how  much  ? 

4.  Divide  f  by  16. 

Note,  When  the  dtvisor  is  a  composite  number,  the  in- 
telligent pupil  will  perceive,  that  he  can  first  divide  by  one 
component  part,  and  the  quotient  thence  arising  by  the 
other ;  thus  he  may  frequently  shorten  the  operation.  In 
the  last  example,  16  =  8x2,  and  f  -^  8  ir:  ^,  and  ^  -r-  2 
=  tV-  ^^^'  tV 

5.  Divide  -^  by  12.     Divide  /^  by  21.     Divide  ff  by  24. 

6.  If  6  bushels  of  wheat  cost  $  ^J  what  is  it  per  bushel  ? 
Note.     The  mixed  number  may  evidently  be  reduced  to 

an  improper  fraction,  and  divided  as  before. 

Ans,  Jl  =  it  of  a  dollar,  expressing  the  fraction  in  its 
lowest  terms.   (TT  46.) 

^  7.  Divide  $  4^^  by  9.  Quot.  ^  oi  ^  dollar. 

^  8.  Divide  12f  by  5.  QuoU  -^=:2f 

9.  Divide  14f  by  8,  QiMt.  Ifj. 

10.  Divide  184^  by  7,  Am.  26^^. 
Note.     When  the  mixed  number  is  Utrge,  it  vnii  be  most 

convenient,  first,  to  divide  the  whole  number,  and  then  re^ 
duce  the  remainder  to  an  improper  fraction ;   and,  after  di- 
viding, annex  the  quotient  of  the  fraction  to  the  quotient  of 
K 


110  FRACTIONS.  IT  49, 50. 

the  ^vhole  number;  thus,  in  the  last  example, dividing  184J 
"by  7,  as  in  whole  numbers,  we  obtain  26  integers,  with  2J 
■=.  ^  remainder,  which,  divided  by  7,  gives  i%  and  26  -f- A 
==  26^^,  Ans. 

11.  Divide  27S6|  by  6.  Ans,  464f. 

12.  How  many  times  is  24  contained  in  7646 J-^  ? 

Aiiii.  318ffJ. 

13.  How  many  times  is  3  contained  in  462 J  ? 

Ans,  154^. 

To  multiply  afraclion  by  a  whole  number. 

^  50.  i.  If  1  yard  of  cloth  cost  -J  of  a  dollar,  what  will 
2  yards  cust  ?  ^  X  2  zi=  how  much  ? 

2.  If  a  cow  consume  j  of  a  bushel  of  meal  in  1  day,  how 
much  will  she  consume  in  3  days  ?    ^  X  3  izi  hov/  much  ? 

.3.  A  boy  bought  5  cakes,  at  f  of  a  dollar  each  \  what  did 
he  give  for  the  whole  ?    f  X  5  zn  how  much? 

4.  How  much  is  2  times  ^  ?  3  times   ^  ?    2 

times  I? 

5.  Multiply  f  by  3. f  by  2.  -J  by  7. 

6.  If  a  man  spend  f  of  a  dollar  per  day,  how  much  will 
he  spend  in  7  days  ? 

f  is  3  parts.  If  he  spend  3  such  parts  in  1  day,  he  will 
evidently  spend  7  times  3,  that  is,  \^-  z_-  2|  in  7  days. 
Hence,  we  perceive,  a  fraction  is  muliiplied  by  multiplying  the 
numerator^  ivilhout  changing  the  denominator. 

But  it  has  been  made  evident,  (1[  49,)  that  multiplying  the 
denominator  produces  the  same  effect  on  the  value  of  the  frac- 
tion, as  dhndlng  the  numerator:  hence,  also,  dividing  the  de- 
nominator will  produce  the  same  effect  on  the  value  of  the 
fraction,  as  midtiplyinn  the  numerator.  In  all  cases,  therefore, 
where  one  of  the  terms  of  the  fraction  is  to  be  multiplied^  the 
same  result  will  be  effected  by  dividing  the  ofhrr ;  and  where 
one  tprm  is  to  be  diirided^  the  same  result  may  be  effected  by 
multiplying  the  other. 

This  principle,  borne  distinctly  in  mind,  will  frequently 
enable  the  pupil  to  shorten  the  operations  of  fractions.  Thus, 
in  the  following  example  : 

At  -^^  of  a  dollar  for  1  pound  of  sugar,  what  will  11  pounds 
cost  ? 

Multiplying  the  numerator  by  11,  we  obtain  for  the  pro- 
duct II  =  I  of  a  dollar  for  the  answer. 


If  61.  FRACTIONS.  Ill 

IT  51<  But,  by  applying  the  above  principle,  and  dividing 
tlie  denominator,  instead  of  multiplijing  the  numerator^  we  at 
once  come  to  the  answer,  |,  in  its  lowest  ternris.  Hence, 
there  are  two  ways  to  muUiply  a  fraction  by  a  zvhole  number  :-^ 

I.  Divide  the  denominator  by  the  whole  number,  (when  it 
can  be  done  witfiout  a  remainder,)  and  over  the  quotient 
write  the  numerator. — Otherwise, 

II.  Multiply  the  numerator  by  the  whole  number,  and  un- 
der the  product  write  the  denominator.  If  then  it  be  an 
improper  fraction,  it  may  be  reduced  to  a  whole  or  mixed 
number. 


EXAMPLES    FOR   PRACTICE. 

1.  If  1  man  consume  -3^^^  of  a  barrel  of  liour  in  a  month, 

how  much  will  18  men  consume  in  the  same  time  ?  —  6 

men  ?  9  men  ?  Ans.  to  the  last^  \\  barrels. 

2.  What  is  the  product  of  fVj  multiplied  by  40  ?  ^^J^  X 
40  r=  how  much  ?  "  Ans.  23|. 

3.  Multiply  ^W  by  12. by  18.  by  21.  by 

36.  by  *48.  by  60. 

Note,  When  the  multiplier  is  a  composite  number,  the 
pupil  will  recollect,  (IT  11,)  that  he  may  first  multiply  by 
one  component  part,  and  that  product  by  th*^  other.  Thus, 
in  the  last  example,  the  multiplier  60  is  equal  to  12  X  5; 
therefore,  j^'^X  12  ziz  -{4^  and  J-J  X  5 1=  -f  |  =:  5^^-,    Ans, 

\  4.  Multiply  5J  by  7.  An.    40J. 

Nnfp.  It  ia  evident,  thnt  the  mixed  number  may  be  re- 
duced to  an  improper  fraction,  and  multiplied,  as  in  the  pre- 
ceding examples  ;  but  the  operation  will  usually  be  shorter, 
to  multiply  the  fraction  and  whole  number  separately^  and 
add  the  results  top^ether.  Thus,  in  the  last  example,  7  times 
5  are  35;  and  7  times  J  are  ^5^=:5|,  which,  addjd  to  35, 
make  40,{^,  Ans. 

Or,  we  may  multiply  the  friction  first,  and,  writing  down 
the  fraction,  reserve  the  integers,  to  be  carried  to  the  product 
of  the  whole  number. 

«.  6,  What  will  9^1}  tons  of  hay  come  to  at  $  17  per  ton  } 

Am,   $  1642\j. 
6.  If  a  man  travel  2£jj  miles  in  1  hour,  how  far  will  he 


112  FRACTIONS.  IT  61, 52S. 

travel  in  5  hours  ? in  8  hours  ?  — -—  in  12  hours  ?  

in  3  days,  supposing  he  travel  12  hours  each  day? 

An»,  to  the  last,  77f  miles. 
Note.     The  fraction  is  here  reduced  to  its  lowest  terms ; 
the  same  w^ill  be  done  in  all  following  examples. 

To  multiply  a  whole  number  by  a  fraction. 

11  52.  1.  If  36  dollars  be  paid  for  a  piece  of  cloth,  what 
costs  f  of  it  ?     36  X  |-  =^  how  much  ? 

f  of  the  quantity  will  cost  |  of  the  price ;  f  a  time  36  dol- 
lars, that  is,  f  of  36  dollars,  implies  that  36  be  first  divided 
into  4  equal  parts,  and  then  that  1  of  these  parts  be  taken  3 
times;  4  into  36  goes  9  times,  and  3  times  9  is  27. 

Ans.  27  dollars. 

From  the  above  example,  it  plainly  appears,  that  the  ob^ 
ject  in  mulliplying  by  a  fraction^  whatever  may  be  the  multipli- 
cand^ is,  to  take  out  of  the  midtiplicand  a  part^  denoted  by  the. 
multiplying  fraction;  and  that  this  operation  is  composed  of 
two  others,  namely,  a  division  by  the  denominator  of  the 
DfHiltiplying  fraction,  and  a  multiplication  of  the  quotient  by 
the  numerator.  It  is  matter  of  indifference,  as  it  respects 
the  residtj  which  of  these  operations  precedes  the  other,  for 
36  X  3  ~  4  =  27,  the  same  as  36  -r-  4  X  3  zn  27. 

Hence, — To  multiply  by  a  fraction^  whether  the  multiplicand 
be  a  whole  number  or  a  fraction^ — 

RULE. 

Divide  the  multiplicand  by  the  denominator  of  the  multi- 
plying fraction,  and  multiply  the  quotient  by  the  numerator ; 
or,  (which  will  often  be  found  more  convenient  in  practice,) 
first  multiply  by  the  numerator,  and  divide  the  product  by 
the  denominator. 

Multiplication,  therefore,  when  applied  to  fractions,  does 
not  always  imply  augmentation  or  increase,  as  in  w^hole 
numbers ;  for,  when  the  multiplier  is  less  than  unityy  it  will 
always  require  the  product  to  be  less  than  the  multiplicandy 
to  which  it  would  be  only  equal  if  the  multiplier  were  1 . 

We  have  seen,  (IT  10,)  that,  when  two  numbers  are  multi- 
plied together,  either  of  them  may  be  made  the  multiplier, 
w^ithout  affecting  the  result.     In  the  last  example,  therefore, 
instead  of  multiplying  16  by  f ,  we  may  multiply  J  by  16 
(TT  60,)  and  the  result  will  be  the  same. 


^62,53.  FRACTIONS.  113 

EXAMPLES    FOR    PRACTICE. 

^2.  What  will  40  bushels  of  corn  come  to  at  f  of  a  dollar 
per  bushel  ?     40  X  f  =  how  much  ? 

3-  What  will  24  yards  of  cloth  cost  at  f  of  a  dollar  per 
yard  ?     24  X  §  ==  how  much  ? 

4.  How  much  is  J-  of  90  ?  f  of  369  ?  ^^.  of  45  ? 

j  5.  Multiply  45  by  y^^.     Multiply  20  by  J-. 

To  muttiplij  one  fraction  by  another. 

TT  53.  1.  A  man,  owning  f  of  a  ticket,  sold  f  of  his 
share ;  what  part  of  the  whole  ticket  did  he  sell  ?  §  of  ^  is 
how  much  ? 

We  have  just  seen,  (IT  52,)  that,  to  multiply  by  a  fraction, 
is  to  divide  the  multiplicand  by  the  denominator^  and  to  rnulti- 
ply  the  quotient  by  the  numerator.  ^  divided  by  3,  the  de- 
nominator of  the  multiplying  fraction,  ('^49,)  is  -^r--,  which, 
multiplied  by  2,  the  numerator,  (11  51,)  is  -j-\,  Ans. 

The  process,  if  carefully  considered,  will  be  found  to  con- 
sist in  multiplying  together  the  two  numerators  for  a  new  mir 
merator^  and  the  two  denominators  for  a  nevj  denom'utulor. 

EXAMPLES    FOR   PRACTICE. 

2.  A  man,  havi.ng  I  o^  ^  dollar,  gave  f  of  it  for  a  dinner ; 
what  did  the  dinner  cost  him  ?  Ans.  J-  dollar. 

.  3.  Multiply  I  by  f .     Multiply  fj  by  f .  Product^  ^^. 

I,  4.  How  much  is  |  of  f  of  J  of  J  ? 

Note.  Fractions  like  the  above,  connected  by  the  word 
ofy  are  sometimes  called  compound  fractions.  The  word  OF 
unplies  their  continual  multiplication  into  each  other. 

Ans.  iU  =  ^(T' 

When  there  are  several  fractions  to  be  multiplied  con- 
tinually together,  as  the  several  numerators  are  factors  of  the 
new  numerator,  and  the  several  denominators  are  factors  of 
the  new  denominator,  the  operation  may  be  shortened  h^ 
dropping  those  factors  lohich  are  the  same  in  both  termSy  on  the 
principle  explained  in  If  46.  Thus,  in  the  last  example,  ^j 
§,  1^,  I,  we  lind  a  4  and  a  3  both  among  the  numerators  and 
among  the  denominators;  therefore  we  drop  them,  multiply- 
ing together  only  the  remaining  numerators,  2  X  '^  ^=^  l'^?  ^^^ 
a  new  numerator,  and  the  remaining  denominators,  5X8  = 
40,  for  a  new  denominator,  making  +J-  =  J^,  Am,  as  before. 


114  FRACTIONS.  IT  63, 54. 

5.  f  of  I  of  f  of  f  of  Y^j  of  f  of  I  =  how  much  ?  Am,  ^5^. 
f  6.  What  is  the  continual  product  of  7,  ^,  ^  of  f  and  3^  ? 

Note,  The  integer  7  may  be  reduced  to  the  form  of  an 
improper  fraction  by  writing  a  unit  under  it  for  a  denomina- 
tor, thus,  -J.  Ans.  2-f|. 

7.  At  ^\  of  a  dollar  a  yard,  what  will  J^  of  a  yard  of  cloth 
cost  ? 

8.  At  6f  dollars  per  barrel  for  flour,  what  will  -f-^  of  a  bar- 
rel cost  ? 

6|-  :=  \i- ;  then  V^  X  t\  =  ^^1=$  2^^,  ■^^^' 

9.  At  |-  of  a  dollar  per  yard,  what  cost  7f  yards  ? 

Ans,    $6^i. 
^  10.  At  $  2^  per  yard,  what  cost  Gf  yards  ?    Am,  $  14f  f . 

II.  What  is  the  continued  product  of  3,  f ,  f  of  f ,  2f ,  and 
i4off  oft  ?  Ans,  fff. 

TT  54.  TAc  Rule  /or  the  multiplication  of  fractions  may 
71010  be  presented  at  one  view  : — 

I.  To  multiply  a  fraction  by  a  whole  number^  or  a  whole 
number  by  a  fraction^ — Divide  the  denominator  by  the  whole 
liumber,  when  it  can  be  done  without  a  remainder;  other- 
wise, midtiply  the  numerator  by  it,  and  under  the  product 
write  the  denominator,  which  may  then  be  reduced  to  a 
whole  or  mixed  number. 

XL  To  multiply  a  mixed  number  by  a  whole  number^ — Multi- 
ply the  fraction  and  integers,  separately^  and  add  their  pro- 
ducts together. 

III.  To  multiply  one  fraction  by  another^ — Multiply  together 
the  numerators  for  a  new  numerator,  and  the  denominators  for 
a  new  denominator. 

Note,  If  either  or  both  are  mixed  numbers^  they  may  fir€t 
be  reduced  to  improper  fractions, 

exampi.es  for  practice. 

1.  At  $f  per  yard,  what  cost  4  yards  of  cloth  ?  5 

yds.  >  6  yds.  ?  8  yds.  }  20  yds.  ? 

Ans.  to  the  last,  $  15. 

2.  Multiply  148  by  i,  -—by  f by  ^. by  ^. 

Last  product,  44^. 

3.  If  2^  tons  of  hay  keep  1  horse  through  the  winter^ 


IT  54j  55.  FRACTIONS.  116 

how  much  will  it  take  to  keep  3  horses  the  same  time?  . 

7  horses  ?  13  horses  ?  Ans.  to  last^  '^7-^  tons. 

4.  What  will  S^^  barrels  of  cider  come  to,  at  $  3  per 
barrel  ? 

5.  At  $  14f  per  cwt.,  what  will  be  the  cost  of  147  cwt.? 
/  6.  A  owned  f  of  a  ticket ;  B  owned  -f-^  of  the  same  ;  the 
ticket  was  so  lucky  as  to  draw  a  prize  of  $  1000  ;  what  was 
each  one's  share  of  the  money  ? 

7.  Multiply  1  of  f  by  f  of  f .  Product,  -J. 

8.  Multiply  7i  by  2^^.  Product,  15^. 

9.  Multiply  i  by  2f .  Product,  2|. 

10.  Multiply  f  of  6  by  |.  Product,  1. 
*  11.  Multiply  f  of  2  by  -^  of  4.  Product,  3. 

12.  Multiply  continually  together  ^  of  8,  f  of  7,  f  of  9, 
and  I  of  10.  Product,  20. 

13.  Multiply  1000000  vy  f .  Product,  555555f. 

To  divide  a  whole  number  by  a  fraction^ 

IT  65.  We  have  already  shown  (IT  49)  how  to  divide  a 
fraction  by  a  whole  number ;  we  now  proceed  to  show  how 
to  divide  a  whole  number  oy  a  fraction. 

1.  A  man  divided  $9  among  some  poor  people,  giving 
them  J  of  a  dollar  each ;  how  many  were  the  persons  who 
received  the  money  ?     9  -^  f  z=:  how  many  ? 

1  dollar  is  |-^  and  9  dollars  is  9  times  as  many,  that  is,  -^ ; 
then  J  is  contained  in  -^/-  as  many  times  as  3  is  contained 
in  36.  Ans.  12  persons. 

That  is, — Multiply  the  dividend  by  the  denominator  of  the 
dividing  fraction,  (thereby  reducing  the  dividend  to  parts  of 
the  same  magnitude  as  the  divisor,)  and  divide  the  product 
by  the  numerator, 

2.  How  many  times  is  |  contained  in  8  ?  8  -j-  f  =  how 
many  ? 

OPERATION. 

8  Dividend. 
5  Denominator^ 

Numerator,  3)40 

Quotient,  13^  times,  the  Answer, 
To  multiply  by  a  fraction,  we  have  seen,  (!!  52,)  implied 
two  operations — a  division  and  a  multiplication;  so,  also,  to 
divide  by  a  fraction  implies  two  operations — a  inultiplication 
and  a  division. 


116 


FRACTIONS. 


IT  59. 


IT  56.    Division  is  the  reverse  of  muUiplication. 


To  multiply  by  a  fraction, 
whether  the  multiplicand  be 
a  whole  number  or  a  fraction, 
as  has  been  already  shown, 
(TT  52,)  we  divMe  by  the  de- 
nominator of  the  multiplying 
fraction,  and  midiiply  the  quo- 
tlent  by  the  numerator. 

Note,  In  either  case,  it  is  matter  of  indifference,  as  it 
respects  the  result,  which  of  these  operations  precedes  the 
other;  but  in  practice  it  will  frequently  be  more  convenient, 
that  the  multiplication  precede  the  division. 


To  divide  by  a  fraction, 
whether  the  dividend  be  a 
whole  number  or  a  fraction, 
we  multiply  by  the  denomina- 
tor of  the  dividing  fraction, 
and  divide  the  product  by  the 
numerator. 


12  multiplied  by  |-,  the  pro- 
duct is  9. 

In  multiplication,  the  mul- 
tiplier being  less  than  unity, 
or  1,  will  require  the  product 
to  be  less  than  the  multipli- 
cand, (If  52,)  to  which  it  is 
only  equal  when  the  multi- 
plier is  1,  and  greater  when 
the  multiplier  is  more  than  1. 


12  divided  by  J,  the  quo- 
tient is  16. 

In  division,  the  divisor  be- 
ing less  than  unity,  or  1,  will 
be  contained  a  greater  number 
of  times;  consequently  will  re- 
quire the  quotient  to  be  great- 
er than  the  dividend,  to  which 
it  will  be  equal  when  the  di- 
visor is  1,  and  less  when  the 
divisor  is  more  than  1. 


exampl.es  for  practice* 

1.  How  many  times  is  ^  contained  in  7  ?  7  -r-  ^  =  bow 
many  ? 

2.  How  many  times  can  I  draw  J  of  a  gallon  of  wine  out 
of  a  cask  containing  26  gallons  ? 

3.  Divide  3  by  J.  6  by  |.  10  by  f. 

4.  If  a  man  drink  y^g^  of  a  quart  of  rum  a  day,  how  long 
will  3  gallons  last  him  ? 

6.  If  2J  bushels  of  oats  sow  an  acre,  how  many  acres  will 
22  bushels  sow  ?     22  -7-  2f  =  how  many  times  ? 

Note.  Reduce  the  mixed  number  to  an  improper  frac- 
tion, 2f  =  \K  Ans,  8  acres. 

6.  At  $4f  a  yard,  how  many  yards  of  cloth  may  be 
bought  for  $  37  ?  Ans.  S/g-  yards. 

7.  How  many  times  is  -f^r^  contained  in  84  ? 

Ans,  90J  times. 


it  66,  67.  FRACTIONS.  lit 

/  8.  How  many  times  is  ^  contained  in  6  ? 

Ans.  f  of  1  time. 

9.  How  many  times  is  8|  contained  in  53 : 

Ans,  6^  times. 

10.  At  f  of  a  dollar  for  building  1  rod  of  stone  wall,  how 
many  rods  may  be  built  for  $  87  ?  87  ~  f  r=  how  many 
times  ? 

To  divide  one  fraction  by  another. 

^  57.  1.  At  §  of  a  dollar  per  bushel,  how  much  rye  may 
be  bought  for  f  of  a  dollar  ?  f  is  contained  in  f  how  many 
times  ? 

Had  the  rye  been  2  whole  dollars  per  bushel,  instead  of  f 
of  a  dollar,  it  is  evident,  that  f  of  a  dollar  must  have  been 
divided  by  2,  and  the  quotient  would  have  been  -^^ ;  but  the 
divisor  is  3ds,  and  3ds  will  be  contained  3  times  where  a 
like  number  of  whole  ones  are  contained  1  time;  conse- 
quently the  quotient  fV  is  3  times  too  smallj  and  must  there- 
fore, in  order  to  give  the  true  answer,  be  multiplied  by  3, 
that  is,  by  the  deiiominator  of  the  divisor; 3  times  t%  =i 
i^^bush*  Ans. 

The  process  is  that  already  described,  IT  55  and  IT  56.  If 
carefully  considered,  it  will  be  perceived,  that  the  numerator 
of  the  divisor  is  multiplied  into  the  denominator  of  the  divi- 
dend, and  the  denominator  of  the  divisor  into  the  numerator 
of  the  dividend  ;  wherefore,  in  practice,  it  will  be  more  con- 
venient to  invert  the  divisor ;  thus,  §  inverted  becomes  i} ; 
then  multiply  together  the  two  upper  terms  for  a  numerator,  and 
the  two  lower  term^  for  a  denominator ^  as  in  the  multiplication 
of  one  fraction  by  another.  Thus,  in  the  above  example, 
^X3_  9 

2'X5-10'^^^^^^'^- 

EXAMPLES    FOR   PRACTICE. 

2.  At  J  of  a  dollar  per  bushel  for  apples,  how  many  bush- 
els may  be  bought  for  |^  of  a  dollar  ?  How  many  times  is  ^ 
contained  in  J  ?  Ans.  3  J  bushels. 

3.  If  |-  of  a  yard  of  cloth  cost  f  of  a  dollar,  what  is  that 
per  yard?  It  will  be  recollected,  (If  24,)  that  when  the  cost 
of  any  quantity  is  given  to  find  the  price  of  a  unit,  w^e  divide 
the  cost  by  the  quantity.  Thus,  f  (the  cost)  divided  by  J 
(the  quantity)  will  give  the  price  of  I  yard. 

Ans.  If  of  a  dollar  per  yard. 


118  FRACTIONS.  IT  57, 59; 

Proof.  If  the  work  be  right,  (TT  16,  "  Proof,")  the  pro- 
duct of  the  quotient  into  the  divisor  will  be  equal  to  the 
dividend  ;  thus,  ft  X  5  =  f •  This,  it  will  be  perceived, 
is  multiplying  the  price  of  one  yard  (f|)  by  the  quantity  (J) 
to  find  the  cost  (f ;)  and  is,  in  fact,  reversing  the  question, 
thus,  If  the  price  of  1  yard  be  f-|  of  a  dollar,  what  will  |  of  a 
yard  cost  ?  Ans.  f  of  a  dollar. 

Note.  Let  the  pupil  be  required  to  reverse  and  prove  the 
succeeding  examples  in  the  same  manner. 

4.  How  many  bushels  of  apples,  at  ^^  of  a  dollar  per 
bushel,  may  be  bought  for  ^  of  a  dollar  ?  Ans.  4|  bushels. 
J  5.  If  4f  pounds  of  butter  serve  a  family  1  week,  how 
many  weeks  will  36 1  pounds  serve  them  ? 

The  mixed  numbers,  it  will  be  recollected,  may  be  re- 
duced to  improper  fractions.  Aiis.  Syg^-  weeks. 

6.  Divide^  by  ^.    Qnot.l.         Divide^  by  ^.  Quot.  2. 

7.  Divide  f  by  i.   Quot.  3.     /  Divide  |  by  j%.       Quot.  f  |» 

8.  Divide  2^  by  U.  Divide  10|  by  2f 

Quot.  IJ.  Quot.  4|^. 

9.  How  many  times  is  j\j  contained  in  f  ?     Ans.  4  times. 

10.  IIow  many  times  is  f  contained  in  4§  ? 

Alts.  11§  times. 

11.  Divide  g  of  J  by  J  of  -f .  Quot.  4. 

^  68.  The  Rule  /t-r  dicision  effractions  may  now  he  pr&* 
seated  at  one  view  : — 

I.  To  divide  a  fraction  by  a  whole  number, — Divide  the 
numerator  by  the  whole  number,  when  it  can  be  done  with- 
out a  remainder,  and  under  the  quotient  write  the  denomi- 
nator ;  otherwise,  multiphj  the  denominator  by  it,  and  over  the 
product  write  the  numerator. 

II.  To  divide  a  ivhole  number  by  a  fraction, — Multiply  the 
dividend  by  the  denominator  of  the  fraction,  and  divide  the 
product  by  the  numerator.^ 

III.  To  divide  one  fraction  by  another, — Invert  ihe  divisor^ 
and  multiply  together  the  two  upper  terms  for  a  numerator, 
and  the  two  lower  terms  for  a  denominator. 

Note.  If  either  or  both  are  mixed  numbers,  they  may  be 
reduced  to  improper  u  actions. 


IT  59.      ADDITION  AND  SUBTRACTION  OF  FRACTIONS.      119 


EXAMPLES   FOR  PRACTICE. 

1.  If  7  lb.  of  sugar  cost  -^^^  of  a  dollar,  what  is  it  pe? 
pound  ?      T^jy  -f-  7  =  how  much  ?     f  of  y^^y  is  how  much  ? 

2.  At  $  1^  for  f  of  a  barrel  of  cider,  what  is  that  per  bar- 
rel? 

3.  If  4  pounds  of  tobacco  cost  J  of  a  dollar,  what  does  1 
pound  cost? 

4.  If  |-  of  a  yard  cost  $  4,  what  is  the  price  per  yard  ? 

5.  If  14f  yards  cost  $  75,  what  is  the  price  per  yard  ? 

Ans.  5^3. 

6.  At  31 J  dollars  for  IQj-  barrels  of  cider,  what  is  that 
per  barrel  ?  Ans.  $  3. 

7.  How  many  times  is  |  contained  in  746  ?      Ans.  1989^. 

8.  Divide  ^  of  f  by  f .  Divide  J  by  f  of  f . 

QuoL  |.  Quot.  3|f . 

9.  Divide  J  of  ^  by  ^  of  f .  Quot.  ^. 

10.  Divide  -J  of  4  by  ^^.  Quot.  3. 

11.  Divide  4f  by  |  of  4.  Quot.  2^. 

12.  Divide  |  of  4  by  4|-.  Quot.  f  ^ 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 

tr  59.  1.  A  boy  gave  to  one  of  his  companions  f  of  an 
orange,  to  another  f,  to  another  -J;  what  part  of  an  orange 
did  he  give  to  all  ?         1  + 1-  +  i  —  l^o^v  much  ?        Ans.  f . 

2.  A  cow  consumes,  in  1  month,  y\  of  a  ton  of  hay ;  a 
horse,  in  the  same  time,  consumes  y\-  of  a  ton  ;  and  a  pair 
of  oxen,  -^ ;  how  much  do  they  all  consume  ?    how  much 

piore  does  the  horse  consume  than  the  cow  ?     the  oxen 

than  the  horse  ?  -f-^  +  y\  +  -j\  =:  how  much  ?  -^  —  A  = 
how  much  ?     -^  —  ^  =  ^^^^  much  ? 

3-  i  +  f +i  =  lio^vmuch  ?     f  —  ;i-  =  howmuch? 

4.  2V+/^  +  ^a  +  M  +  2^(T  =  h0WmUch?      il  — T^ff  = 

how  much  ? 

5.  A  boy,  having  J  of  an  apple,  gave  ^  of  it  to  his  sister; 
what  part  of  the  apple  had  he  left  ?     f  —  i=:  how  much  ? 


120       ADDITION  AND  SUBTRACTION  OF  FRACTIONS.     U  60. 

When  the  denominators  of  two  or  more  fractions  are  alikey 
(as  in  the  foregoing  examples,)  they  are  said  to  have  a  common 
denominator.  The  parts  are  then  in  the  same  denomina- 
tion, and,  consequently,  of  the  same  magnitude  or  value.  It 
is  evident,  therefore,  that  they  may  be  added  or  subtracted, 
by  adding  or  subtracting  their  numerators,  that  is,  the  num- 
ber of  their  parts,  care  being  taken  to  write  under  the  re- 
sult their  proper  denominator.    Thus,  ^-f-ir^^ifj  i  —  f 

6.  A  boy,  having  an  orange,  gave  -J-  of  it  to  his  sister,  and 
^  of  it  to  his  brother ;  what  part  of  the  orange  did  he  give 
away  ? 

4ths  and  8ths,  being  parts  of  different  magnitudes,  or  value, 
cannot  be  added  together.  We  must  therefore  first  reduce 
them  to  parts  of  the  same  magnitude,  that  is,  to  a  common  de- 
nmninaior.  f  are  3  parts.  If  each  of  these  parts  be  divided 
into  2  equal  parts,  that  is,  if  we  multiply  both  terms  of  the 
fraction  f  by  2,  (IT  46,)  it  will  be  changed  to  f ;  then  f  and 
J  are  J.  Ans,  |^  of  an  orange, 

7.  A  man  had  f  of  a  hogshead  of  molasses  in  one  cask, 
and  f  of  a  hogshead  in  another;  how  much  mt>ie  in  one 
cask  than  in  the  other  ? 

Here,  3ds  cannot  be  so  divided  as  to  become  5ths,  nor 
can  5ths  be  so  divided  as  to  become  3ds  ;  but  if  the  3ds  be 
each  divided  into  5  equal  parts,  and  the  5ths  each  into  3 
equal  parts,  they  will  all  become  15ths.  The  f  will  become 
^^,  and  the  |-  will  become  -^ ;  then,  -f-^  taken  from  -J-J  leaves 

IT  60.  From  the  very  process  of  dividing  each  of  the 
parts,  that  is,  of  increasing  the  denominators  by  multiplying 
them,  it  follows,  that  each  denominator  must  be  di  factor  of 
the  common  denominator  ;  now,  multiplying  all  the  denomina- 
tors together  will  evidently  produce  such  a  number. 

Hence,  To  reduce  fractions  of  different  denominators  to 
equivalent  fractions,  having  a  common  denominator, — ^RuLE  : 
Multiply  together  all  the  denominators  for  a  common  denotni' 
nator,  and,  as  by  this  process  each  denominator  is  multiplied 
by  all  the  others,  so,  to  retain  the  value  of  each  fraction, 
multiply  each  numerator  by  all  the  denominators,  except  it» 
own,  for  a  new  numerator,  and  xind'cr  it  write  the  common 
denominator. 


V  00.    ADDITIOKT  ANU  SUBTRACTION  OF  FRACnOLiS.        121 

EXAMPLES   FOR  PRACTICE, 

1.  Reduce  §,  J  and  ^  to  fractions  of  equal  value,  having  a 
common  denominator. 

3  X  4  X  5  n=  60,  the  common  denominator. 
2  X  4  X  5  =  40,  the  new  numerator  for  the  iirst  fraction. 
3X3X5  =  45,  the  new  numerator  for  the  second  fraction. 
3X4X4  =  48,  the  new  numerator  for  the  third  fraction. 

The  new  fractious,  therefore,  are  |{;,  fj,  and  ||.  By  an 
inspection  of  the  operation,  the  pupil  will  perceive,  that  the 
numerator  and  denominator  of  each  fraction  have  been  mul- 
tiplied by  the  same  numbers;  consequently,  (IT  46,)  that 
their  value  has  not  been  altered.  ^  ^    ^     ' 

2.  Reduce  ^,  f,  ^  and  f  to  equivalent  fi€cfioni^,  having' .. 
common  denominator.  Ans.  jf  g-,  -}f  J,  J|3-,  ^^g, 

3.  Reduce  to  equivalent  fractions  of  a  common  denomi- 
nator, and  add  together,  ^,  f ,  and  ^. 

^ns.  1-5  +  If  +  i^  =  f  J  =  Hh  Amount, 
74.  Add  together  J  and  f .  Amount^  1  J|. 

\  5.  What  is  the  amount  of  ^  -f-  ^  -f  1  +  -^  ? 

Arts,  m=}AV 

6.  What  are  the  fractions  of  a  common  denominator 
equivalent  to  J  and  f  ?  Ans.  ^f  and  f £,  or  -fir  and  |J. 

We  have  already  seen,  (^  59,  ex.  7,)  "that  the  common 
denominator  may  be  any  number,  of  which  each  given  de- 
nominator is  a  factor,  that  is,  any  number  which  may  be  di- 
vided by  each  of  them  without  a  remainder.  Such  a  number 
is  called  a  common  multiple  of  all  its  common  divisors,  and 
the  least  number  that  \vi\\  do  this  is  called  their  least  com- 
mon multiple;  therefore,  the  least  common  denominator  o^  imy 
fractions  is  the  least  common  multiple  of  all  their  denominators. 
Though  the  rule  already  given  will  always  find  a  common 
multiple  of  the  given  denominators,  yet  it  will  not  always 
find  their  least  comrao'/  multiple.  In  the  last  example,  24 
is  evidently  a  common  multiple  of  4  and  6,  for  it  will  exactly 
measure  both  of  them;  but  12  will  do  the  same,  and  as  12  is 
the  least  number  that  will  do  this,  it  is  the  least  common 
multiple  of  4  and  6.  It  will  theiefore  be  convenient  to  have 
A  rule  for  finding  this  least  common  multiple.  Let  the  num- 
bers be  4  and  6. 

It  is  evident,  that  on^  number  is  a  multiple  of  another, 
when  the  former  contains  all  the  factors  of  the  latter.     The 
L 


122         ADDITIOIV   AlfD  iSUBTKACTIOrf  OF  FRACTIONS.     IT  61, 

factors  of  4  are  2  and  2,  (2  X  2  =  4.)  The  factors  of  6 
are  2  and  3,  (2  X  3  z=  6.)  Consequently,  2  X  2  X  3  =  12 
contains  tlie  factors  of  4,  that  is,  2  X  2  ;  and  also  contains  the 
factors  of  6,  th<tt  is,  2X3.  12,  then,  is  a  common  multiple 
of  4  and  6,  and  it  is  the  least  common  multiple,  because  it 
does  not  contain  any  factor^  except  those  whkh  make  up 
the  numbers  4  and  6 ;  nor  either  of  those  repeated  more 
than  is  necessary  to  produce  4  and  6.  Hence  it  follows, 
that  when  any  two  numbers  have  a  factor  common  to  both, 
it  may  be  once  omitted ;  thus,  2  is  a  factor  common  both  to 
4  and  6,  and  is  consequently  once  omitted. 

IT  61,  On  this  principle  is  founded  the  Rule  for  finding 
the  least  conmon  multiple  of  two  or  more  numbers.  Write 
down  the  numbers  in  a  line,  and  divide  them  by  any  number 
that  will  measure  two  or  more  of  them  ;  and  write  the  quo- 
tients and  undivided  numbers  in  a  line  beneath.  Divide  this 
line  as  before,  and  so  on,  until  there  are  no  two  numbers 
that  can  be  measured  by  the  same  divisor ;  then  the  conti- 
nual product  of  all  the  divisors  and  numbers  in  the  last  line 
will  be  the  least  common  multiple  required. 

Let  us  apply  the  rule  to  find  the  least  common  multiple 
of  4  and  6. 

V  4  and  6  may  both  be  measured  by  2 ;    the 

f      '  quotients  are  2  and  3.    There  is  no  number  great- 

2  .  3  er  than  1,  which  will  measure  2  and  3.  There- 
fore, 2X2X3i=:12is  the  least  common  mul- 
tiple of  4  and  6. 

If  the  pupil  examine  the  process,  he  will  see  that  the  di- 
visor 2  is  a  factor  common  to  4  and  6,  and  that  dividing  4 
by  this  factor  gives  for  a  quotient  its  other  factor,  2.  In  the 
same  manner,  dividing  6  gives  its  other  factor,  3.  Therefore 
the  divisor  and  quotients  make  up  all  the  factors  of  the  two 
numbers,  which,  mullifilied  together,  must  give  the  com- 
mon multiple. 

7.  Reduce  f ,  ^,  f  and  ^  to  equivalent  fractions  of  the 
least  common  denominator. 

OPERATION.  Then,  2x^X2=12,  least  common 

^  )  ^  '  ^  *  3  .  6  denominator.  It  is  evident  we  need 
3)2.1.3.3  not  multiply  by  the  Is,  as  this  would 
2  ,  \  ,  \  ,  \       ^^^  ^^^"^  ^^^^  number. 


1r  61.     ADDITION  AND  SUBTRACTION  OF  FRACTIONS.       123 

To  find  the  new  numerators,  that  is,  how  many  12tbs 
each  fraction  is,  we  may  take  f ,  ^,  |  and  ^  of  12.     Thus  : 

f  of  12  =  9  \  /  A.  =z  f 

iqC  12 6  /  ^^^^""^Grators,  which,  i  _£_ * 

2    f  1 9  ZI  o  /  written   over   the  com-  /  \^  ~  f 
^  ^.  \  mon  denominators,  ffive  /  ^^  —  ^ 

^ofl2  =  2)  ""        (A  =  i 

-*  Ans.  ^%  -iv,  ^8^  and  -^. 

I  8.  Reduce  ^,  |,  and  |^  to  fractions  having  the  least  com- 
inoa  denominator,  and  add  them  together. 

Ans,  if  +  ^\  +  §i  =  i-i  ==  lih  amount. 

9.  Reduce  ^  and  ^  to  fractions  of  the  least  common  de- 
nominator, and  subtract  one  from  the  other. 

Ans,  ^^  —  -/^  z=z  yig^,  dilFerence. 

10.  What  is  the  least  number  that  3,  5,  8  and  10  will 
measure?  Ans,  120. 

11.  There  are  3  pieces  of  cloth,  one  containing  7|  yards, 
another  13|  yards,  and  the  other  15|  yards;  how  many 
yards  in  the  3  pieces. 

Before  addii.g,  reduce  the  fractional  parts  to  their  least 
common  denominator;  this  being  done,  we  shall  have, 

^    Adding  together  all  the  24ths,  viz.  18  -f-  20 

Z      t     }      +21'  ^^^   obtain    69,   that  is,   M  =  2^J- 

13f  =  13f  J  V      \ye  write  down  the  fraction  J-^  under  the 

15|-  =  lof^-  )      other  fractions,  and  reserve  the  2  iu.egers 

A       .3^  J  J  to  be  carried  to  the  amount  of  the  other 

^^*  integers,  making  in  the  whole  37^^,  Ans, 

12.  There  was  a  piece  of  cloth  containing  34f  yards, 
from  which  were  taken  12|  yards ;  how  much  was  there 
left  ? 

—  OA  9  ^^^  cannot  take  16  twenty-fourths 

t """  (it)  ^^^^  ^  twenty-fourths,  {^\  ;)  we 

12f  rz:  124f  must,  therefore,  borrow  1  integer,  =ii  24 

^n,.iT«yt&.  twenty-fourths,  (f|,)  which,  with /,., 

^*  ^  makes  Jl ;  we  can  now  take  ^J  irom 

ff ,  and  there  will  remain  ^ ;  but,  as  we  'borrowed,  so  also 

we  must  carry  1  to  the  12,  which  makes  it  13,  and  13  from 

34  leaves  21.  Ans.  21  J|. 

13.  What  is  the  amount  of  j-  of  f  of  a  yard,  f  of  a  yard, 
and  "t  of  2  yards  ? 

Note,  The  compound  fraction  may  be  reduced  to  a  siwi 
pie  fraction ;  thus,  ^  of  J  =:  f  ;  and  -J  of  2  =  f ;  then,  J  4 


124  EEDUCTIOX    OF   FRACTIONS.  IT  62,  63^ 

IT  62.  From  the  foregoing  examples  we  derive  the  fol- 
lowing Rule  : — To  add  or  subtract  fractions^  add  or  subtract 
their  numerators^  wh-jin  they  have  a  common  denominator  \ 
otherwise,  they  must  first  be  reduced  to  a  commou  denomi- 
nator. 

Note,  Compound  fractions  must  be  reduced  to  simple 
fractions  before  adding  or  subtracting. 

EXAMPLES    FOR   PRACTICE. 

1.  What  is  the  amount  off,  4f  and  12?  Arts.  17^. 

2.  A  man  bought  a  ticket,  and  sold  |  of  J  of  it ;  what  part 
of  the  ticket  had  he  left  ?  Am,  f . 

3.  Add  together  J,  |,  ^,  /^,  -J  and  ^J.  Amount^  2^. 
^4.  What  is  the  difference  between  14^j  and  16^  ? 

Am.  1^. 

5.  From  1^  take  |.  Remainder^  f. 

6.  From  3  take  i-  Rem.  2f. 

7.  From  147 J  take  48f .  Rem.  98J. 

8.  From  I  of  ^  take  ^  of  ■^.  Rein.  ^^. 

9.  Add  together  112^,  31  If,  and  lOOOJ-. 

10.  Add  together  14,  11,  4f,  j^  and  i. 

11.  From  f  take  ^.     From  J  take  f. 

12.  What  is  the  difference  between  ^  and  i?  f  and  i'f 
Jandf?     fandf?     fandf?     ^  and  J  ? 

13.  How  much  is  1— i?  1—4?  1  — ft  ?  1  — f  ? 
2-f?     2^4^?     2f~f?     3t  — ^V.'     1000  — tV-? 


REDUCTION  OF  FRACTIONS. 

IT  63.  We  have  seen,  (TT  33,)  that  integers  of  one  denomi- 
nation may  be  redticed  to  integers  of  another  denomination* 
It  is  evident,  that  fractions  of  one  denomination,  after  the 
same  manner,  and  by  the  same  rides,  may  be  reduced  to 
fractions  of  another  denomination.;  that  is,  frocliom,  like 
integers,  may  be  brought  into  lower  denominations  by  mul^ 
tiplicaXiqn,  and  into  higher  denominations  by  division. 


T63. 


REDUCTION    OF    FRACTIONS. 


125 


To  reduce  Jtigher  into  lower 

denominations. 

(Rule.     See  IT  34.) 

I.  Reduce  rrl-^  of  a  pound 
to  pence,  cr  the  fraction  of  a 
penny. 

Note,  Let  it  be  recollect- 
ed, that  a  fractijn  is  multiplied 
either  by  dividing  its  denomi- 
nator^ or  by  multiphjing  its  nu- 
merator, 

^-Ijj  iS  X  20  =  yV  s.  X  12 
=  |-  d.  Ans. 

Or  thus:  ^^^  oi  23.  oi  if- = 
f  |§  =  f  of  a  penny,  Ans, 

3.  Reduce  xsVtf  ^^  ^  pound 
to  the  fraction  of  a  farthing. 

X5-Vzy£.X20z:z^f|-,S.  X 
12=:T¥8"aJ.  X4zz:-,^^Vo  = 

Or  thus  : 
Num,  1 

20  s.  in  1  £, 

20 

12  d.  in  1  s. 

240 

4  q.  in  1  d. 

960 

Then,  T¥A~Sq-  ^ris, 
6.  Reduce  2-/g-5  of  a  guinea 
to  the  fraction  of  a  penny. 
f  7.  Reduce  f  of  a  guinea  to 
the  fraction  of  a  pound. 
Consult  ir  34,  ex.  11. 
9.  Reduce  |  of  a  moidore,  at 
36  s.  to  the  fraction  of  a  guinea. 

II.  Reduce  ^j  of  a  pound, 
Troy,  to  the  fraction  of  an 
ounce. 


To  reduce  lower  into  higher 
denominations, 

(Rule.    See  1134.) 
2.  Reduce  f  of  a  penny  to 
the  fraction  of  a  pound. 

Note,  Division  is  perform- 
ed either  by  dividing  the  «w- 
meratorj  or  by  mxdliphjing  the 
denominator, 

f  d.  -4-  12  —  ^x^- 8.-^20  = 
■2i^£,  Ans. 

Or  thus :  f  of  ^^  of  -^^  = 
-6— —  -4.-^.  Ans. 


■12-m ' 


4.  Reduce  f  of  a  farthing 
to  the  fraction  of  a  pound. 

J  q.  --  4  izz  W  d.  -M2  = 
_|^s.--20z=:^Vtj^^^^^. 

Or  thus  : 
Denom,  4 

4  q.  m  1  a. 

16 

12  d.  in  1  s. 

192 
20  s.  inl  £. 


3840 
Then,  j^ffV^: 


■T2 


^£.Am. 


6.  Reduce  |  of  a  penny  to 
the  fraction  of  a  guinea. 

8.  Reduce  -|  of  a  pound  to 
the  fraction  of  a  guinea. 

/lO.  Reduce  f  J  of  a  guinea 
to  the  fraction  of  a  moidore. 

12.  Reduce  J  of  an  ounce 
to  the  fraction  of  a  pound 
Troy, 


126 


REDUCTION    OF    FRACTIONS.  IT  63,  64, 


13.  Reduce  ^^g^  of  a  pound, 
avoirdupois,  to  the  fraction  of 
an  ounce. 

f  lo,  A  man  has  y^^  of  a 
hogshead  of  wine;  what  part 
is  that  of  a  pint  ? 

#  17.  A  cucumber  grew  to  the 
length  of  -j^VfJ  ^^  ^  ^^^^  5  what 
part  is  that  of  a  foot  ? 

^19.  Reduce  f  of  i  of  a 
pound  to  the  fraction  of  a  shil- 
ling. 

21.  Reduce  |  of  ^-j-  of  3 
pounds  to  the  fraction  of  a 
penny. 


IT  6<i,  it  will  frequently  be 
required  to  find  the  value  of  a 
fraclion^  that  is,  to  reduce  a 
fracHon  to  integers  of  less  de- 
nominations, 

1.  What  is  the  value  of  f 
of  a  pound?  In  other  words. 
Reduce  f  of  a  pound  to  shil- 
lings and  pencQ. 

f  of  a  poimd  is  4|X  =  13  J  shil- 
lings ;  it  is  evident  from  -J  of 
a  shilling  may  be  obtained 
some  pence  ;  -J  of  a  shilling  is 
Af  =  4  d.  That  \s,— Multiply 
the  numerator  by  that  number 
which  will  reduce  it  i*^  the  next 
less  dznomination^  and  divide 
the  product  by  the  denominator; 
if  there  he  a  remainder^  multiply 
and  divide  as  before^  and  so  on  ; 
the  several  quotients^  placed  one 
after  another^  in  their  order^ 
wUl  be  the  answer. 


*  14.  Reduce  f  of  an  ounce 
to  the  fraction  of  a  pound 
avoirdupois. 

16.  A  man  has  y^j  of  a  pint 
of  wine ;  what  part  is  that  of 
a  hogshead  ? 

■  18.  A  cucumber  grew  to 
the  length  of  1  foot  4  inches 
z=  -i|  z=  ^  of  a  foot ;  what  part 
is  that  of  a  mile  ? 

20.  f  ^  of  a  shilling  is  f  of 
what  fraction  of  a  pound  ? 

^22.  JL8_o.  of  a  penny  is  -J  of 
what  fraction  of  3  pounds  ? 
-L8jO  of  a  penny  is  ^  of  what 
part  of  3  pounds  ?  ^^  of  a 
penny  is  J  of  ^  of  how  many 
pounds  ? 

It  will  frequently  be  re- 
quired to  reduce  integers  to 
the  fraction  of  a  greater  de- 
nomination. 

2.  Reduce  13  s.  4  d.  to  the 
fraction  of  a  pound. 

13  s.  4  d.  is  160  pence; 
there  are  240  pence  in  a 
pound ;  therefore,  13  s.  4  d.  is 
^f  0  _.  I  of  a  pound.  That 
is, — Reduce  the  given  svm  or 
quantity  to  the  least  denomina- 
tion mentioned  in  it,  for  a  nu- 
merator ;  then  reduce  an  inte- 
ger of  that  greater  denomina-^ 
tion  (to  a  fraction  of  which  it 
is  required  to  reduce  the  giv- 
en sum  or  quantity)  to  the 
smne  denomination,  for  a  denomi* 
nator,  and  they  will  form  the 
fraction  required* 


IT  64. 


REDUCTION    OF    FRACTIONS. 


121 


EXAMPLES  FO>l  PRACTICE 

3.  What  is  the  value  of  f 
of  a  shilling  ? 

OPERATION. 
Numer,       3 
12 


Den(m,S)Ze(4  d.  2  q. 
32 

4 
4 


Am, 


9.  |-  of  a  month  is  howma- 
hy  days,  hours,  and  minutes  ? 


11.  Reduce  f  of  a  mile 
its  proper  quantity; 


to 


^'  13.  Reduce  -{^  of  an   acre 
A6  its  proper  quantity. 
^X  15.  What  is  the  value  of 
•fj   of  a   dollar   in   shillings, 
pence,  &c.  ? 

17.  What  is  the  value  of  ^j- 
of  a  yard  ? 

19.  What  is  the  value  of  y% 
of  a  ton  ? 


EXAMPLES  FOR  PRACTICE. 

4.  Reduce  4  d.  2  q.  to  the 
fraction  of  a  shilling. 

OPERATION. 
4  d.  2  q.  Is. 

4  12 

18  Numcr.  12 

4 

48  Denotth 
if  =  I  Ans. 


16(2  q. 
16 

5.  What  is  the  value  of  f    '  6.  Reduce  7  oz.  4  pwt.  to 
of  a  pound  Troy  ?  the  fraction  of  a  pound  Troy. 

/  7.  What  is  the  value  off      8.  Reduce  8  oz.  14f  dr.  to 
of  a  pound  avoirdupois  .''  the  fraction  of  a  pound  avoir- 

dupois. 

Note.  Both  the  numerator 
and  the  denominator  must  be 
reduced  to  9ths  of  a  dr. 

10.  3  weeks,  1  d.  9  h.  36  m. 
is  what  fraction  of  a  month  ? 

12.  Reduce  4  fur.  125  yds. 
2  ft.  1  in.  2f  bar.  to  the  frac- 
tion of  a  mile. 

14.  Reduce  1  rood  30  poles 
to  the  fraction  of  an  acre. 

16.  Reduce  5  s.  7^  d.  to 
the  fraction  of  a  dollar. 

18.  Reduce  2  ft  8  in.  1|  b. 
to  the  fraction  of  a  yard. 

20.  Reduce  4  cwt.  2  qr.  12 
lb.  14  oz.  12y\-  dr.  to  the  frac- 
j  tion  of  a  ton* 

Note,     Let  the  pupil  be  required  to  rex^erse  and  prove  the 
following  examples : 

21.  What  is  the  value  of  -^^  of  a  guinea? 


128  SUPPLEMENT    TO    FRACTIONS.  IT  64, 

22.  Reduce  3  roods  17^^  poles  to  the  fraction  of  an  acre. 
'23.  A  man  bought  27  gal.  3  qts.  1  pt.  of  molasses;  what 
part  is  that  of  a  hogshead  ? 

/  24.  A  man  purchased  -f*^  of  7  cwt.  of  sugar;  how  much 
isugar  did  he  purchase  ? 

25.   13  h.  42  m.  51f  s.  is  what  part  or  fraction  of  a  day  ? 


SUPPIiEMENT  TO   FRACTIONS.  i 

QUESTIONS. 

1.  What  2iXQ  fractions  1  2.  Whence  is  it  that  the  parts 
into  which  any  thing  or  any  number  may  be  divided,  take 
their  name  ?  3.  How  are  fractions  represented  by  figures  ? 
4.  What  is  the  number  above  the  line  called  ? — Why  is  it 
so  called  ?  5.  What  is  the  number  below  the  line  called  ?• 
— Why  is  it  so  called  ? — Wliat  does  it  show?  6.  What  is 
it   which  determines  the  magnitude  cf  the  parts? — Why? 

7.  What  is  a  simple  or  proper  fraction?  an  improper 

fraction  ?  a  mixed  number  ?     8.  How  is  an  improper 

fraction  reduced  to  a  whole  or  mixed  number?     9.  How  is 

a  mixed  number  reduced  to  an  improper  fraction  ?  a 

whole  number?  10.  What  is  understood  by  the  terms  of  the 
fraction?  11.  How  is  a  fraction  reduced  to  its  most  simple 
or  lowest  terms  ?  12.  W^hat  is  understood  by  a  common  di- 
visor?   by  the  greatest  common  divisor?     13.   How  is 

it  found  ?  14.  How  many  ways  are  there  to  multiply  a  frac- 
tion by  a  whole  number?  15.  How  does  it  appear,  that  di- 
viding the  denominator  multiplies  the  fraction  ?  16.  How  is  a 
mixed  wwmhex  multiplied?  17.  What  is  implied  in  multi- 
plying by  a  fraction  ?  18.  Of  how  many  operations  does  it 
consist? — What  are  they?  19.  When  the  multiplier  is  less 
than  a  unit,  what  is  the  product  compared  with  the  multi- 
plicand ?  20.  How  do  you  multiply  a  whole  number  by 
a  fraction?  21.  How  do  you  multiply  one  fraction  by  ano- 
ther? 22.  How  do  you  multiply  a  mixed  number  by  a 
mixed  number  ?  23.  How  does  it  appear,  that  in  multiply- 
ing both  terms  of  the  fraction  by  the  same  number  the  value 
of  the  fraction  is  not  altered  ?  24.  How  many  ways  are 
there  to  divide  a  fraction  by  a  whole  number  ? — What  are' 
they?  25.  How  does  it  appear  that  diffraction  is  divided  by 
multiplying  its  denwninatorl     26.  How  does  dividing  by  ft 


1r  64,  65.  SUPPLEMENT   TO    FRACTIONS.  J  2d 

fraction  differ  from  multiplying  by  a  fraction  ?  27.  When 
the  divisor  is  less  Iban  a  unit,  what  is  the  quotient  compared 
with  the  dividend  ?     28.  What  is  understood  by  a  common 

denominator  ?  the   least   common    denominator  ?     29. 

How  does  it  appear,  that  each  given  denominator  mi:st  be  a 
factor  of  the  common  denominator  ?  30.  How  is  the  com- 
mon denominator   to   two   or   more  fractions   found?     3L 

What  is  understood  by  a  multiple  ?  by  a  common  miilti" 

plel  by  the  least  common  multiple  ? — What  is  the  pro- 
cess of  finding  it  ?  32.  How  are  fractions  added  and  sub- 
tracted ?     33.  How  is  a  fraction  of  a  greater  denomination 

reduced  to  one  of  a  less?  of  a  less  to  a  greater?     34. 

How  are  fractions  of  a  greater  denomination  reduced  to  in- 
tegers of  a  less  ?  integers  of  a  less  denomination  to  the 

fraction  of  a  greater  ? 


EXERCISES, 

1.  What  is  the   amount   of  |  and  |  ?  of  ^  and  f  ? 

of  12^,  3§  and  4  J  ?  Ans.  to  the  last,  20-^- 

2.  To  |-  of  a  pound  add  f  of  a  shill_^ng.         Amount,  18^  s. 
Note,     First  reduce  both  to  the  same  denomination. 

3.  |-  of  a  day  added  to  J  of  an  hour  make  how  many 
hours  ?  what  part  of  a  day  ?  Ans,  to  the  last,  f  f  d- 

4.  Add  ^  lb.  Troy  to  f^-  of  an  oz. 

Amount,  6  oz.  11  pwt.  16  ^. 
•  5.  How  much  is  i  less  ^?     t%  — i-^     t^  — /*j^     ^H 
—  4^?     6-~4t?     if?  — ioff  off? 

Am.  to  the  last,  ^4?. 

6.  From  J  shilling  take  J  of  a  penny.  Rem,  5^  d. 

7.  From  ^  of  an  ounce  take  J  of  a  pwt. 

Rem,  11  pwt.  3  grs, 

8.  From  4  days  7^  hours  take  1  d.  Q^V  h. 

Rem,  2  d.  22  h.  20m. 

9.  At  $^  per  yard,  what  costs  J  of  a  yard  of  cloth  ? 

IT  65.  The  price  of  unity,  or  1,  being  given,  to  find  the 
cost  of  any  quantity,  either  less  or  more  than  uuity,  multiply 
the  price  by  the  quantity.  On  the  other  hand,  the  cost  of  any 
quantity,  either  less  or  more  than  unity,  being  given,  to  find 
the  price  of  unity,  or  1,  divide  the  cost  by  the  quantity, 

Ans.   $  ^ 


It 

l30  SUPPLEMENT    TO    FRACTIONS.  IT  65. 

1.  If -fj  lb.  of  sugar  cost  -^-^  of  a  shilling,  what  will  Jf  of 
a  pound  cost  ?* 

This  example  will  require  two  operations  :  first,  as  above, 
to  find  the  price  of  1  lb. ;  secondly,  having  found  the  price 
of  1  lb.,  to  find  the  cost  of  f f  of  a  pound,  t^  s.  -7-  -}^  (-ff 
of  f^s.  IT  57)  nr  y9^V  s-  the  price  of  1  lb.  Then,  ^^^  s.  X 
.  B  (If  of  tU  s.  II  53)  ^  m-i  s.  =  4  d.  3f?H  q.,  the 
Answer, 

Or  we  rnay  reason  thus  :  first  to  find  the  price  of  1  lb. : 
\^  lb.  costs  y^3-  s.  If  we  knew  what  -^  lb.  would  cost, 
we  might  repeat  this  13  times,  and  the  result  would  be  the 
price  of  1  lb.  \^  is  11  parts.  If  ^^  lb.  costs  ^  s.,  it  is  ce- 
dent jV  1^'  will  cost  ^j-  of  f"^  r=  yj-j-  s.,  and  if  lb.  will  cost 
13  times  as  much,  that  is,  -f^-^  s.  =  the  price  of  1  lb.  Then, 
H  of  AV  s.  =  'iUl  s.,  the  cost  of  e  of  a  pound,  f  ^f  s, 
rr:  4  d.  3f  fl^^  q.,  as  before.  This  process  is  called  solving 
the  question  by  analysis. 

After  the  same  manner  let  the  pupil  solve  the  following 
questions : 

2.  If  7  lb.  of  sugar  co?;t  f  of  a  dollar,  what  is  that  a 
pound  ?  4  of  f  =  how  much  ?  What  is  it  for  4  lb.  ?  ^ 
of  f  =^  how  much  ?  What  for  12  pounds  ?  y  of  f  zr  how 
much  ?  Ans,  to  the  last,  $  If. 

3.  If  6  J  yds.  of  cloth  cost  $  3,  what  cost  9|-  yards  ? 

Ans.    $4'269. 

4.  If  2  oz.  of  silver  cost  $  2^24,  what  costs  f  oz.  ? 

f.  Ans.    $*84. 

*  5.  If  ^  oz.  costs  $\i,  what  costs  1  oz.  ?  Ans.  $  1'283. 
i  6.  If  1^  lb.  less  by  ^  costs  13-J  d.,  what  costs  14  lb.  less  by 
h  of  2  lb.  ?  .         A71S.  4£ .  9  s.  9^  d. 

7.  Iff  yd.  cost  $1,  what  will  40J  yds.  cost? 

Am.    $59^062+. 

8.  If  y^g-  of  a  ship  costs  $  251,  what  is  -^  of  her  worth  ? 

.  Ans.    $53^785+. 

\  9.  At  3|£.  per  cwt.,  what  will  9f  lb.  cost? 

Ans.  6  s.  3^d. 

10.  A  merchant,  owning  ^  of  a  vessel,  sold  §  of  his  share 
for  $957 ;  what  was  the  vessel  worth  ?     Ans.    $  1794*375. 

11.  Iff  yds.  cost  ^£-,  what  will  -f^  of  an  ell  Eng.  cost? 

Ans.   17  s.  1  d.  2f  q. 

•  This  and  the  followins^  are  examples  usually  referred  to  theTuIe  Proportion, 
or  Rule  of  Three,    See  1[  95  ex.  35, 


tr  65.  SUPPLEMENT   TO    FRACTIONS.  131 

^  12.  A  merchant  bought  a  number  of  bales  of  velvet,  each 
containing  129^  yards,  at  the  rate  of  $7  for  5  yards,  and 
sold  them  out  at  the  rate  of  $  11  for  7  yards,  and  gained 
$  200  by  the  bargain  ;  how  many  bales  were  there  ? 

First  find  for  what  he  sold  5  yards ;  then  what  he  gained 
on  5  yards — what  he  gained  on  1  yard.  Then,  as  many  times 
as  the  sum  gained  on  1  yd.  is  contained  in  $  200,  so  many 
yards  there  must  have  been.  Having  found  the  number  of 
yards,  reduce  them  to  bales.  Am.  9  bales. 

^  13.  If  a  staff,  5f  ft.  in  length,  cast  a  shadow  of  6  feet,  how 
high  is  that  steeple  whose  shadow  measures  153  feet? 

Aris.  144-J-  feet. 
,  14.  If  16  men  finish  a  piece  of  work  in  28  J  days,  how 
'long  will  it  take  12  men  to  do  the  same  work  ? 

First  find  how  long  it  would  take  1  man  to  doit ;  then  12 
men  will  do  it  in  ^  of  that  time.  Ans.  37|-  days. 

15.  How  many  pieces  of  merchandise,  at  20^-  s.  apiece, 
must  be  given  for  240  pieces,  at  12^1-  s.  apiece  ?   Ans,  149^^^ 

16.  How  many  yards  of  bocking  that  is  1|-  yd.  wide  will 
be  sufficient  to  line  20  yds.  of  camlet  that  is  f  of  a  yard 
wide  ? 

First  find  the  contents  of  the  camlet  in  square  measure ; 
then  it  will  be  easy  to  find  how  many  yards  in  length  of 
bocking  that  is  1 J  yd.  wide  it  Avill  take  to  make  the  same 
quantity.  A71S,  12  yards  of  camlet. 

"t"  17.  If  1|-  yd.  in  breadth  require  20^  yds.  in  length  to 
make  a  cloak,  what  in  length  that  is  J  yd.  wide  will  be  re- 
quired to  make  the  same  ?  Ans,  34^  yds. 

18.  If  7  horses  consume  2f  tons  of  hay  in  6  weeks,  how 
many  tons  will  12  horses  consume  in  8  weeks  ? 

If  we  knew  how  much  1  horse  consumed  in  1  week,  it 
would  be  easy  to  find  how  much  12  horses  would  consume 
in  8  weeks. 

2|.  =  -Lt  tons-  If  7  horses  consume  ^  tons  in  6  weeks, 
1  horse  will  consume  |  of  Jji^  =  4^  of  a  ton  in  6  weeks  ;  and 
if  a  horse  consume  ^  J  of  a  ton  in  6  weeks,  he  will  consume 
^  of  a  z=  ^g.  of  a  ton  in  1  week.  12  horses  will  consume 
12  timos  yV^  =z  -fff  in  1  week,  and  in  8  weeks  they  will 
consume  8  times  -iff  =::  -^-f-  z=i  6f  tons,  Ans, 
\  19.  A  man  with  his  family,  which  in  ail  were  5  persons, 
did  usually  drink  7^  gallons  of  cid'^r  in  1  week;  how  much 
will  they  drink  in  22^  weeks  when  3  persons  more  are 
added  to  the  family  ?  Ans.  280|  gallons. 


13^  DECIMAL   FHAi^riONS.  IT  66, 67* 

^^  20.  If  9  students  spend  lOJiB.  iu  18  days,  liow  much 
will  20  students  spend  in  30  days  ?     Aiis.  39^2 .  18  s.  4ff  d. 


If  S6,  We  have  seen,  that  an  individual  thing  or  number 
may  be  divided  into  any  number  of  equal  parts,  and  that 
these  parts  will  be  called  halves,  thirds,  fourths,  fifths,  sixths, 
&c.,  according  to  the  number  of  parts  into  which  the  thing 
or  number  may  be  divided  ;  and  that  each  of  these  parts  may 
be  again  divided  into  any  other  number  of  equal  parts,  and  so 
on.  Such  are  called  common^  or  vulgar  fractions.  Their  denom- 
inators are  not  uniform,  but  vary  with  every  varying  division 
of  a  unit.  It  is  this  circumstance  which  occasions  the  chief 
difficulty  in  the  operations  to  be  performed  on  them ;  for 
when  numbers  are  divided  into  different  kinds  or  parts,  they 
cannot  be  so  easily  compared.  This  difficulty  led  to  the  in- 
vention of  decimal  fractions,  in  which  an  individual  thing,  or 
number,  is  supposed  to  be  divided  first  into  te7i  equal  parts, 
which  will  be  tenths  ;  and  each  of  these  parts  to  be  again  di- 
vided into  ten  other  equal  parts,  which  will  be  hundredths ; 
and  each  of  these  parts  to  be  still  further  divided  into  ten 
other  equal  parts,  which  will  be  thousandths ;  and  so  on. 
Such  are  called  decimal  fractions,  (from  the  Latin  worddeceniy 
which  signifies  ten,)  because  they  increase  and  decrease,  in  a 
tenfold  proportion,  iu  the  same  manner  as  whole  numbers. 

IT  67.  In  this  way  of  dividing  a  unit,  it  is  evident,  that 
the  denominator  to  a  decimal  fraction  will  always  be  10, 
100,  1000,  or  i  with  a  number  of  ciphers  annexed;  conse- 
quently, the  denominator  to  a  decimal  fraction  need  not  be 
expressed,  for  the  numerator  only,  written  with  a  point  be- 
fore it  (')  called  the  sepa.ratrix,  is  sufficient  of  itself  to  ex- 
press the  true  value.     Thus, 

-^^     are  written  ^6. 

tVtt   '27. 

^i^  '685. 

TTie  denominator  to  a  decimal  fraction,  although  not  ex- 
pressed, is  always  understood,  and  is  1  with  as  many  ci- 
phers annexed  as  there  are  places  in  the  numerator.  Thus, 
*3765  is  a  decimal  consisting  of  four  places  ;  consequently, 
1  with  four  ciphers  annexed  (10000)  is  its  proper  denomina- 
tor.    Any  decimal  may  be  expressed  in  tlie  form  of  a  conH 


IT  67 


DECIMAL   FRACTIONS. 


183 


mon  fraction  by  writing  under  it  its  proper  denominator. 
Thus,  ^3765  expressed  in  the  form  of  a  common  fraction, 

When  whole  numbers  and  decimals  are  expressed  to- 
gether, in  the  same  number,  it  is  called  a  mixed  number. 
Thus,  25*63  is  a  mixed  number,  25*,  or  all  the  figures  on  the 
left  hand  of  the  decimal  point,  being  whole  numbers,  and 
'63,  or  all  the  figures  on  the  right  hand  of  the  decimal  point, 
being  decimals. 

The  names  of  the  places  to  ten-millionths,  and,  generally, 
how  to  read  or  write  decimal  fractions,  may  be  seen  from 
the  following 

TABIiE. 


CO 

2 


CD 


3d  place. 
2d  place. 
1st  place. 

"  1st  place. 
2d  place. 
3d  place. 
4th  place. 
5th  place. 
6th  place. 
7th  place. 


»  II  II  ii  II  II  II  11 


rf^    Ox    *:|   O 


Hundreds. 

Tens. 

Units. 


ooQOOoiOOCn 
o  oa  o  Oi  oo  en  oi 


o 
o 


o  o  en  o 
o  Q  W 
o  fc^ 


Tenths 
Hundredths. 
Thousandths. 
Ten-Thousandths 

Hundred-Thousandths. 

Millionths. 

Ten-Millionthfl. 


M 


O*    bO    •Jl   00 

gp  en  "7  o> 

g^  S:  S* 
Bon 

p. 


o 

0 


Oi  Ox  OX  CJt 

CO  H  c  g 

5  S  S  » 

p  GO  n>  • 

§ 


134  DECIMAL    FRACTIONS.  IT  68,  69. 

From  the  table  it  appears,  that  the  first  figure  on  the  right 
hand  of  the  decimal  point  signifies  so  many  tenth  parts  of  a 
unit;  the  second  figure,  so  many  hundredth  parts  of  a  unit; 
the  third  figure,  so  many  thousandth  parts  of  a  unit,  &c.  It 
takes  10  thousandths  to  make  1  hundredth,  10  hundredths 
to  make  1  tentli,  and  10  tenths  to  make  1  unit,  in  the  same 
manner  as  it  takes  10  units  to  make  1  ten,  10  tens  to  make 
1  hundred,  &c.  Consequently,  we  may  regard  unity  as  a 
starting  point,  from  whence  whole  numbers  proceed,  con- 
tinually increasing  in  a  tenfold  proportion  towards  the  left 
hand,  and  decimals  continually  decreasing^  in  the  same  pro- 
portion, towards  the  right  hand.  But  as  decimals  decrease 
towards  the  right  hand,  it  follows  of  course,  that  they  in- 
crease towards  the  left  hand,  in  the  same  manner  as  whole 
numbers. 

IT  68.  The  value  of  every  figure  is  determined  by  its 
place  from  units.  Consequently,  ciphers  placed  at  the  right 
hand  of  decimals  do  not  alter  their  value,  since  every  signifi- 
cant figure  continues  to  possess  the  same  place  from  unity. 
Thus,  ^5,  '50,  '500  are  all  of  the  same  value,  each  being 
equal  to  ^^g-,  or  J-. 

But  every  cipher,  placed  at  the  left  hand  of  decimal  frac- 
tions, diminishes  them  tenfold,  by  removing  the  significant 
figures  further  from  unity,  and  consequently  making  each 
part  ten  times  as  small.  Thus,  '5,  '05,  '005,  are  of  different 
value,  '5  being  equal  to  y%-,  or  J  ;  '05  being  equal  to  yj^,  or 
2V ;  and  '005  being  equal  to  py%-^-,  or  ^-^-^y. 

Decimal  fractions,  having  different  denominatorSy  are  readily 
reduced  to  a  common  denominator^  by  annexing  ciphers  until 
they  are  equal  in  number  of  places.  Thus,  '5,  '06,  '234  may 
be  reduced  to  '500,  '060,  '234,  each  of  which  has  1000  for  a 
common  denominator. 

^  69.  Decimals  are  read  in  the  same  manner  as  whole 
numbers,  giving  the  name  of  the  lowest  denomination,  or 
right  hand  figure,  to  the  whole.  Thus,  '6S53  (the  lowest 
denomination,  or  right  hand  figure,  being  ten-thousandths) 
is  read,  6853  ten-thousandths. 

Any  whole  number  may  evidently  be  reduced  to  decimal 
parts,  that  is,  to  tenths,  hundredths,  thousandths,  &c.  by  an- 
nexing ciphers.  Thus,  25  is  250  tenths,  2500  hundredths^ 
25000  thousandths,  &c.     Consequently,  any  mixed  number 


IT  70.       ADDITION  AND  SUBTRACTION  OF  DECIMALS.       136 

may  be  read  together,  giving  it  the  name  of  the  lowest  de- 
nomination or  right  hand  figure.  Thus,  25'63  may  be  read 
2563  hundredths,  and  the  whole  may  be  expressed  in  the 
form  of  a  common  fraction,  thus,  ^^Vc^* 

The  denominatfons  in  federal  money  are  made  to  corre- 
spond to  the  decimal  divisions  of  a  unit  now  described,  dol- 
lars being  units  or  whole  numbers,  dimes  tenths,  cents  hun- 
dredths, and  mills  thousandths  of  a  dollar ;  consequently  the 
expression  of  any  sum  in  dollars,  cents,  and  mills,  is  simply 
the  expression  of  a  mixed  number  in  decimal  fractions. 

Forty-six  and  seven  tenths  =.  46^(j  =  46 '7. 

Write  the  following  numbers  in  the  same  manner : 

/    Eighteen  and  thirty-four  hundredths. 
Fifty-two  and  six  hundredths. 

Nineteen  and  four  hundred  eighty-seven  thousandths. 
Twenty  and  forty-two  thousandths. 
One  and  five  thousandths. 
135  and  3784  ten-thousandths. 
9000  and  342  ten-thousandths.  1  UISTI V 

10000  and  15  ten-thousandths.  X^P/^  pa,  «p^  ' 

974  and  102  millionths.  ^''^4s^4k!£i^^ 

320  and  3  tenths,  4  hundredths  and  2  thousandths. 
500  and  5  hundred-thousandths. 
47  millionths. 
Four  hundred  and  twenty-three  thousandths. 


ADDITION    AND    SUBTRACTION    OF    DECIMAL 
FRACTIONS. 

ir  70-  As  the  value  of  the  parts  in  decimal  fractions  in- 
creases in  the  same  proportion  as  units,  tens,  hundreds,  &:c., 
and  may  be  read  together^  in  the  same  manner  as  ^vhole 
numbers,  so,  it  is  evident  that  all  the  operations  on  decimal 
fractions  may  be  performed  in  the  same  manner  as  on  whole 
numbers.  The  only  difficulty,  if  any,  that  can  arise,  must 
be  in  finding  ichere  to  place  the  decimal  point  in  the  result. 
This,  in  addition  and  subtraction,  is  determined  by  the  same 
rule ;  consequently,  they  may  be  exhibited  together. 

1.  A  man  bought  a  barrel  of  flour  for  $  8,  a  firkin  of  but- 


1S6       ADDITION  AND  SUBTRACTION  OF  DECIMALS.       IT  tO. 

ter  for  $  3^50,   7  pounds  of  sugar  for  Sd^  cents,  an  ounce 
of  pepper  for  6  cents  ;  what  did  he  give  for  the  whole  ? 

OPERATION. 
$  8'        =    8000  mills,  or  lOOOths  of  a  dollar.     * 
3*50    =    3500  mills,  or  lOOOths. 
^835  =      835  mills,  or  lOOOths. 
*06    =        60  mills,  or  lOOOths. 

Ans.    $  12'395  =  12395  mills,  or  lOOOths. 

As  the  denominations  of  federal  money  correspond  with 
the  parts  of  decimal  fractions,  so  the  rules  for  adding  and 
subtracting  decimals  are  exactly  the  same  as  for  the  same 
operations  in  federal  money.     (See  IT  28.) 

2.  A  man,  owing  $375,  paid  $175*75;  how  much  did 
he  then  owe  ? 

OPERATION. 
$  375'      z=  37500  cents,  or  lOOths  of  a  dollar. 
175*75  =  17575  cents,  or  lOOths  of  a  dollar. 

$  199*25  =:  19925  cents,  or  lOOths. 

The  operation  is  evidently  the  same  as  in  subtraction  of 
federal  money.  Wherefore, — In  th€  addition  and  subtrac- 
tion of  decimal  fractions, — Rule  :  Write  the  numbers  under 
each  other,  tenths  under  tenths,  hundredths  under  hun- 
dredths, according  to  the  value  of  their  places,  and  point  off 
in  the  results  as  many  places  for  decimals  as  are  equal  to  the 
greatest  number  of  decimal  places  m  any  of  the  given  num- 
bers. 

EXAMPLES   FOR   PRACTICE. 

^ 

3.  A  man  sold  wheat  at  several  times  as  follows,  viz. 
13*25  bushels;  8*4  bushels;  23*051  bushels,  6  bushels,  and 
*75  of  a  bushel;  how  much  did  he  sell  in  the  whole  ? 

Ans,  51*451  bushels. 

4.  What  is  the  amount  of  429,  21^^^,  355y^^,  l^hr  and 
1-,^  ?  Ans,  808^^^,  or  808*143. 
■^  5.  What  is  the  amount  of  2  tenths,  80  hundredths,  89 
thousandths,  6  thousandths,  9  tenths,  and  5  thousandths  ? 

Ans,  2. 
6«  What  is  the  amount  of  three  hundred  twenty-nine,  and 
seven  tenths  ;  thirty-seven  and  one  hundred  sixty-two  Uiou- 
sandths,  and  sixteen  hundredths  ? 


IF  to,  71.      mul'TiplicAtiox  of  decimals.  137 

7.  A  man,  owing  $43165  paid  $376^865;  how  much  did 
be  then  owe?  Am.    $3939436. 

8.  From  thirty-five  thousand  take  thirty-five  thousandths. 
/  Ans,  34999*966. 

9.  From  5'83  take  4*2793.  Ans.  1*5507. 

10.  From  480  take  245*0075.  Ans.  234*9925. 
^  11.  What  is  the  diff^erence  between  1793*13  and  817* 
05693?                                                               Alls.  976*07307. 

12.  From  4Tfg-  take  2^^.  Remainder^  '^-twh^  ^^^  l'^®- 

13.  What  is  the  amount  of  29^^,  374^^^^^^^,  97-^^^%, 
315t(jW)  27,  and  100^%  ?  Ans.  942*957009. 


MULTIPLICATION    OF    DECIMAL    FRACTIONS. 

IT  71.    1.  How  much  hay  in  7  loads,  each  containing 
23*571  cwt.  ? 

OPERATION. 
23*.571  cwt.  =z    23571    lOOOths  of  a  cwt. 
7  7 


Ans.  164*997  cwt.  =1  164997    lOOOths  of  a  cwt. 

We  may  here  (*!T  69)  consider  the  muUiplicand  so  many 
thousandths  of  a  cwt.,  and  then  the  product  will  evidently  be 
thousandths^  and  will  be  reduced  to  a  mixed  or  whole  num- 
ber by  pointing  off  3  figures,  that  is,  the  same  number  as  are 
in  the  multiplicand ;  and  as  either  factor  may  be  made  the 
multiplier,  so,  if  the  decimals  had  been  in  the  multiplier^  the 
same  number  of  places  must  have  been  pointed  off  for  deci- 
mals. Hence  it  follows,  we  must  always  i^oint  ojf  in  the  pro- 
duct as  many  places  for  decimals  as  there  are  decimal  places  in 
both  factors. 

2.  Multiply  *75  by  *25. 

OPERATION.  In  this  ex^ample,  we  have  4  de- 

J^^  cimal  places  in  both  factors ;  we 

^^  must  therefore  point  off  4  places 

3»75  for  decimals  in  the  product.    The 

250  reason  of  pointing  off  this  num- 

^ her  may  appear  still  more  plain, 

U875  Product.  if  we  consider  the  two  factors  as 

M* 


i$d  MULTIPLICATION    OF    DJElCIMALS*  IT  71^ 

common  or  vulgar  fractions.  Thus,  '75  is  -^^j  and  *25  is 
^^s :  now,f^^^  X  xVo"  =  loVo^j  ==  '1875,  Ans.  same  as  b^ 
fore. 

3.  Multiply  '125  by  '03. 

i|25  rlere,  as  the  number  of  significant 

<Q3  figures  in  tlie  product  is  not  equal  to 

the  number  of  decimals  in  both  fac- 

'00375  Prod.  tors,  the  deficiency  must  be  supplied 

by  prefixing  ciphers,  that  is,  placing 
them  at  the  left  hand.  The  correctness  of  the  rule  may  ap- 
pear from  the  following  process :  '125  is  yWjj  ^^^  '^3  is 
yfo- :  now,  yio¥(j  X  t^s  =  ibi>¥uT)  =  '00375,  the  same  as 
before. 

These  examples  v/ill  be  sufficient  to  establish  the  following 
RULE. 

In  the  multiplication  of  decimal  fractions^  multiply  as  in  whole 
numbers  J  and  from  the  product  point  off  so  many  figures  for  deci- 
mals as  there  are  decimal  places  in  the  mxdtiplicand  and  midt'^ 
plier  counted  together^  and^  if  there  are  not  so  many  figures  in  the 
.  product f  supply  the  deficiency  by  jyrefixing  ciphers, 

EXAMPLES    FOR    PRACTICE. 

4.  At  $  5'47  per  yard,  what  cost  8'3  yards  of  cloth  ? 

Ans,    $45'40I. 
f  5.  At  $  '07  per  pound,  what  cost  2G'5  pounds  of  rice  ? 

Ans,    $1'855. 

6.  If  a  barrel  contain  1'75  cwt.  of  flour,  what  will  be  the 
weight  of  '63  of  a  barrel  ?  Ans,  1'1025  cwt. 

7.  If  a   melon   be  worth   $  '09,  what  is  '7  of  a  melon 
wortli  ?  Ans.  6^^  cents. 

8.  Multiply  five  hundredths  by  seven  thousandths. 

Product,  '00035. 

9.  What  is  '3  of  116  ?  Ans.  34'8. 

10.  What  is  '85  of  3672  >  Ans.  3121'2. 

11.  What  is  '37  of '0563?  Ans.  '020831. 

12.  Multiply  572  by  '58.  Product,  331 '76. 
i  13.  Multiply  eighty-six  by  four  hundredths. 

Producty  3'44. 

14.  Multiply  '0062  by  '0008. 

15.  Multiply  forty-seven  tenths  by  one  thousand  eighty- 
six  hundredths. 


^71^72.  DIVISION    OF    DECIMALS.  139 

16.  Multiply  two  hundredths  hy  eleven  thousandths. 

17.  What  will  be  the  cost  of  tliirieeii  huiiJiedths  of  a  ton 
of  hay,  at  $  11  a  ton  ? 

18.  What  will  be  the  cost  of  three  hundred  seventy-five 
thousandths  of  a  cord  of  wood,  at  $  2  a  cord  ? 

J^  19.  If  a  man's  wages  be  seventy-tive  hundredths  of  a  dol- 
lar a  day,  how  much  will  he  earn  in  4  weeks,  Sundays  ex- 
cepted ? 


DIVISION    OF    DECIMAL    FRACTIONS. 

IT  72.  Multiplication  is  proved  by  division.  We  have 
seen,  in  multiplication,  that  the  decimal  places  in  the  product 
must  always  be  equal  to  the  number  of  decimal  places  in  the 
multiplicand,  and  multiplier  counted  together.  The  muki- 
plicand  and  multiplier,  in  proving  multiplication,  become  the 
divisor  and  quotient  in  division.  It  follows  of  course,  in  di- 
vision, that  the  number  of  decimal  places  hi  the  divisor  and 
quotient^  counted  toijether^  must  always  be  equal  to  the  number  of 
decimal  places  in  the  dividend.  This  will  still  further  appear 
from  the  examples  and  illustrations  which  follow : 

1.  If  6  barrels  of  Hour  cost  $44'71S,  what  is  that  a  bar- 
rel ? 

By  taking  away  the  decimal  point,  $44^718  nz  44718 
mills,  or  lOOOths,  which,  divided  by  6,  the  quotient  is  7453 
mills,  zn  $  7^453,  the  Answer, 

Or,  retaining  the  decimal  point,  divide  as  in  whole  num- 
bers. 

OPERATION.  As  the  decimal  places  in  the  di- 

6)44  718  visor  and  quotient,  counted  toge- 

Ans.  7'453  ther,  must  be  equal  to  the  number 

of  decimal  places  in  the  dividend, 
there  being  no  decimals  in  the  divisor^ — therefore  point  off 
three  figares  for  decimals  in  the  quotientj  equal  to  the  number 
of  decimals  in  the  dividend,  which  brings  us  to  the  same  re- 
sult as  before. 

2.  At  $  4^7^  a  barrel  for  cider,  how  many  barrels  may  be 
bought  for  $31  ? 

In  this  example,  there  are  decimals  in  the  divisor,  and 
none  in  the  dividend.  $4'75=i:475  cents,  and  $31,  by 
winexing  two  ciphers,  =  3100  cents ;  that  is,  reduce  the  di 


140  DIVISION   OF   DECIMALS.  H  72. 

vidend  to  parts  of  the  same  denomination  as  the  divisor. 
Then,  it  is  plain,  as  many  times  as  475  cents  are  contained  in 
3100  cents,  so  many  barrels  may  be  bought. 

475) 3100 (6f|^  barrels,  the  Answer;  that  is,  6  barrels  and 
2850  Iff  of  another  barrel. 

-- -—  But  the  remainder,  250,  instead  of  be- 

ing expressed  in  the  form  of  a  common 
fraction,  may  be  reduced  to  lOths  by  annexing  a  cipher, 
which,  in  effect,  is  multiplying  it  by  10,  and  the  division  con- 
tinued, placing  the  decimal  point  after  the  6,  or  whole  ones 
already  obtained,  to  distinguish  it  from  the  decimals  which 
are  to  follow.  The  points  may  be  withdrawn  or  not  from 
the  divisor  and  dividend. 

OPERATION. 
4'75)31'00(6^526  + barrels,  the  Answer;  that  is,  6  bat- 
2850  rels  and  626  tliousandths  of  another 

•— --  barrel. 

^^^^  By  annexing  a  cipher  to  the  first 

remainder,    thereby   reducing    it   to 

1250  lOths,  and  continuing  the  division, 

950  ^^  obtain  from  it  '5,  and  a  still  fur- 

ther  remainder  of  125,  which,  by  an- 

3000  nexing  another  cipher,  is  reduced  to 

2850  lOOths,  and  so  on. 

■~T77  The  last  remainder,  150,  is  ^§  of 

a  thousandth  part  of  a  barrel,  whicli 
is  of  so  trifling  a  value,  as  not  to  merit  notice. 

If  now  we  count  the  decimals  in  the  dividend,  (for  every 
cipher  annexed  to  the  remainder  is  evidently  to  be  counted 
a  decimal  of  the  dividend,)  we  shall  find  them  to  be  five^ 
which  corresponds  with  the  number  of  decimal  places  in  the 
divisor  and  quotient  counted  together. 

3.  Under  IT  71,  ex.  3,  it  was  required  to  multiply  425  by 
'03 ;  the  product  was  '00375.  Taking  this  product  for  a 
dividend,  let  it  be  required  to  divide  '00375  by  '125.  One 
operation  will  prove  the  other.  Knowing  that  the  number 
of  decimal  places  in  the  quotient  and  divisor,  counted  to- 
gether, will  be  equal  to  the  decimal  places  in  the  dividend, 
we  may  divide  as  in  whole  numbers,  being  careful  to  retain 
the  decimal  points  in  their  proper  places.     Thus, 


V  72,  73.  DIVISION   OF   DECIMALS.  141 

OPERATION.  The  divisor,  125,  in  375  goes  3 

425)  *00375  (*03  times,  and  no  remainder.     We  have 

^^^  only  to  place  the  decimal  point  in 

QQQ  the  quotient,  and  the  work  is  done. 

There  are  five  decimal  places  in  the 
dividend  ;  consequently  there  must  be  five  in  the  divisor  and 
quotient  counted  together ;  and,  as  there  are  three  in  the  di- 
visor, there  must  be  two  in  the  quotient ;  and,  since  we  have 
but  one  figure  in  the  quotient,  the  deficiency  must  be  sup- 
plied by  prefixing  a  cipher. 

The  operation  by  vulgar  fractions  will  bring  us  to  the 
same  result  Thus,  425  is  ^2^%,  and  *00375  is  yiAs^  • 
now,  ^'^cu  -^  A%  =  yf K-&S8ir  =  yfu  =  '^3,  the  same 
as  before. 

IT  73.  The  foregoing  examples  and  remarks  are  suffi- 
cient to  establish  the  following 

RUIiE. 

In  the  division  of  decimal  fractions^  divide  as  in  whole  mmi" 
beiSj  and  from  the  right  hand  of  the  qiiotient  point  off  as  many 
figures  for  decimals^  as  the  decimal  figures  in  the  dividend  eX" 
ceed  those  in  the  divisor^  and  if  there  are  not  so  many  figures  in 
the  quotient  J  supply  the  deficiency  by  prefixing  ciphers. 

If  at  any  time  there  is  a  remainder,  or  if  the  decimal 
figures  in  the  divisor  exceed  those  in  the  dividend,  ciphers 
may  be  annexed  to  the  dividend  or  the  remainder,  and  the 
quotient  carried  to  any  necessary  degree  of  exactness ;  but 
the  ciphers  annexed  must  be  counted  so  many  decimals  of 
the  dividend. 

EXAMPLES   FOR   PRACTICE. 

>4.  If  $  472^875  be  divided  equally  between  13  men,  how 
much  will  each  one  receive  ?  A?is,   $  36 '375. 

5.  At  $  '75  per  bushel,  how  many  bushels  of  rye  can  be 
bought  for  $  141  ?  Ans,  188  bushels. 

f  6.  At  12J  cents  per  lb.,  how  many  pounds  of  butter  may 
be  bought  for  $  37  ?  Ans.  296  lb. 

7.  At  6^  cents  apiece,  how  many  oranges  may  be  bought 
for  $8?  Ans.  128  oranges. 

8.  If  '6  of  a  barrel  of  flour  cost  $  5,  what  is  that  per  bar- 
rel? Ans.   $8'333-f^ 

9.  Divide  2  by  534.  Quoi.  ^037+. 


142  REDUCTION    OF   COMMON   OR  Hi  73,  74r 

10.  Divide  ^012  by  *005.  QiioL  2*4. 

II 1.  Divide  three  thousandths  by  four  hundredths. 
^  QuoL  '075, 

'  12.  Divide  eighty-six  tenths  by  ninet}  four  thousandths. 

13.  How  many  times  is  47  contained  in  8  ? 


REDUCTION  OF  COMMON   OR  VULGAR  FRAC- 
TIONS TO  DECIMALS. 

U  74.  1.  A  man  has  |  of  a  barrel  of  flour;  what  is  that 
expressed  in  decimal  paits? 

As  many  times  as  the  denominator  of  a  fraction  is  con- 
tained in  the  numerator,  so  many  whole  ones  are  contained 
in  the  fraction.  We  can  obtain  no  whole  ones  in  |^,  because 
the  denominator  is  not  contained  in  the  numorator.  We 
may,  however,  reduce  the  numerator  to  lentlis^  (IT  72,  ex.  2,) 
by  annexing  a  cipher  to  it,  (wliich,  in  effect,  is  multiplying 
it  by  10,)  making  40  tenths,  or  4^0.  Then,  as  many  times  as 
the  denominator,  5,  is  contained  in  40,  so  many  tenths  are 
contained  in  the  fraction.  5  into  40  goes  8  times,  and  no 
remainder.  Ans,  '8  of  a  bushel. 

2.  Express  f  of  a  dollar  in  decimal  parts. 

The  numerator,  3,  reduced  to  tenths,  is  -Jg,  3^0,  which, 
divided  by  the  denominator,  4,  the  quotient  is  7  tenths,  and 
a  remainder  of  2.  This  remainder  must  now  be  reduced  to 
hundredths  by  annexing  another  cipher,  making  20  hun- 
dredths. Then,  as  many  times  as  the  denominator,  4,  is  con- 
tained in  20,  so  many  hundredths  also  may  be  obtained.  4 
into  20  goes  5  times,  and  no  remainder,  f  of  a  dollar,  there- 
fore, reduced  to  decimals,  is  7  tenths  and  5  hundredths,  that 
is,  '75  of  a  dollar. 

The  operatic  u  may  be  presented  in  form  as  follows  : — 

Num, 
Denom,  4  )  3'0  (^75  of  a  dollar,  the  Answer. 
28 

20 
20 


If  Y4.  VULGAR    FRACTIONS    TO    DECIMALS,  143 

8.  Reduce  ^g^  lo  a  decimal  fraction. 
The  numerator  must  be  reduced  to  hundredths,  by  annex* 
ing  two  ciphers,  before  the  division  can  begin. 

66  )  4*00  (  '0606  +,  the  Ajiswer. 
396 


400  As  there  can  be  no  tenths,  a  cipher  must 

396         be  placed  in  the  quotient,  in  tenth's  place. 

4 

Note.  ^  cannot  be  reduced  exactly;  for,  however  long  the 
division  be  continued,  there  will  still  be  a  remainder.*  It  is 
sufficiently  exact  for  most  purposes,  if  the  decimal  be  ex- 
tended to  three  or  four  places. 

From  the  foregoing  examples  we  may  deduce  the  follow- 
ing general  Rule  : — To  reduce  a  common  to  a  decimal  froA> 

*  Decimal  figures,  which  continually  repeat,  like  *06,  in  this  exam- 
ple, are  called  Rcpetends.,  or  Circulating  Decimals.  If  only  one  figure 
repeats,  as  *3333  or  '•1111 ,  &c.j  it  is  called  a  single  rcpeteud.  If  tzco  ot 
more  figures  circulate  alternately,  as  *06060(i,  '234234234,  &c.,  it  is 
called  a  compound  rcpeteud..  If  other  figures  arise  before  those  which 
circulate,  as  '743333,  '143010101,  &c.,  the  decimal  is  called  a  mixed 
repetend. 

Ji  single  repetend  is  denoted  by  writing  only  the  circulating  figura 
with  a  point  over  it:  thus,  '3,  signifies  that  the  3  is  to  be  continually 
repeated,  forming  an  infinite  or  never-ending  series  of  3's. 

Jl  compound  repetend  is  denoted  by  a  point  over  the  first  and  last  re- 
peating figure :  thus,  '234  signifies  tliat  234  is  to  be  continually  re- 
peated. 

It  may  not  be  amiss,  hero  to  show  how  the  value  of  any  repetend  m^j 
bo  found,  or,  in  other  words,  how  it  may  be  reduced  to  its  equivalent 
vulgar  fraction. 

If  we  attempt  to  reduce  ^  to  a  decimal^  we  obtain  a  continual  repe- 
tition of  tlie  figure  1 :  thus,  'lllll,  that  is,  the  repetejid  '1.  The  value 
of  the  repetend  '1,  then,  is  -^ ;  the  value  of  '222,  &c.,  the  repetend  *2, 
will  evidently  bo  twice  as  much,  that  is,  f-.  In  the  samo  manner,  3=» 
6",  and  '4  =  |,  and  'f»  ==  |-,  and  so  on  to  9,  which  =  f  =  1. 

1.  What  is  the  value  of  '8  .?  Ms.  f . 

2.  What  is  the  value  of  '6  ?  Ans.  f  =  f .  Wliat  is  the  value  of  'G  ? 
of '7? of '4.?    of'o.^    of '9.?    of*i.? 

If  -^  be  reduced  to  a  decimal,  it  produces  'OlOJOl.or  the  repetend  OJ. 
The  repetend  '02,  being  2  times  as  much,  must  be  ^g-  and  '03  =  -jp^p 
aiv]  *48,  being  48  times  a.s  much,  must  be  f  f .  and  '74  ===  Jf ,  (&c. 


144   REDUCT,  OF  VULG.  FRACTIONS  TO  DECIMALS.  IT  74, 

tion, — Annex  one  or  more  ciphers^  as  may  be  necessary,  to  the 
numerator  J  and  divide  it  by  the  denominator.  If  then  there  he 
a  remainder,  annex  another  cipher,  and  divide  as  before,  and  so 
continue  to  do  so  long  as  there  shall  continue  to  be  a  remainder, 
or  until  the  fraction  shall  be  reduced  to  any  necessary  degree 
of  exactness.  The  quotient  will  be  the  decimal  reqviired, 
which  must  consist  of  as  many  decimal  places  as  there  are 
ciphers  annexed  to  the  numerator ;  and,  if  there  are  not  so 
many  figures  in  the  quotient,  the  deficiency  must  be  sup- 
plied by  prefixing  ciphers. 

exampl.es  for  practice. 

4,  Reduce  .V,  J,  -^^-^,  and  y^^  to  decimals. 

Ans,  '5  ;  '25  ;  *025  ;  '00797  +, 

/^5.  Reduce  fj,  ^-^ft^^,  yy\tj  ^^^  ^uV^-^  to  decimals. 

'  Ans.  '692  +  ;  '003;  '0028 +;  '000183 +, 

6.  Reduce  |^,  ^^-q%,  ^^^  to  decimals. 

7.  Reduce  |,  ^V,  ^^,  h  h  tt?  A?  wi^  to  decimals. 

8.  Reduce  i,  f ,  |,  i,  f ,  f ,  f,  ^,  ^\,  '^%  to  decimals. 


If  wh  1>6  reduced  to  a  decimal,  it  produces  '001 ;  consequently, 
'002  =  ef  9 ,  and  '037  =  /^V?  »ind  425  =  ff f,  &c.  As  this  principle 
will  apply  to  any  number  of  places,  we  have  this  general  Rule  for  re- 
ducing a  circulating  decimal  to  a  vulgar  fraction^ — Make  the  given 
repetend  the  numerator,  and  the  denominator  will  be  as  many  9s  as 
there  are  repeating  figures. 

3.  What  is  the  vulgar  fraction  equivalent  to  '704  ?  Ans.  i%i* 

^  4.  What  is  the  value  of  '003  ?    '014  ?    '324  ? 'OlOSi  ? 

"2463  ? '002103  ?  ,ans.  to  last,  TsWinr- 

5.  What  is  the  value  of  '43  ? 

In  this  fraction,  the  repetend  begins  in  the  second  place,  or  place  of 
hundredths.  The  first  figure,  4,  is  X(j,  and  the  repetend,  3,  is  f-  of  -^j 
that  is,  "9^;  these  two  parts  must  be  added  together,  t^^  -f-  ^  =  f^ 
=  -Jtj.  Ans.  Hence,  to  find  the  value  of  a  mixed  repetend^ — Find  the 
value  of  the  two  parts,  separately,  and  add  them  together. 

6.  What  is  the  value  of '153  ?         i^^^T  4"  F^TT  =  iU  =  t¥^7  >^ns. 

7.  What  is  the  value  of  '0047  ?  Ans.  -^Mu* 

8.  What  is  the  value  of '138.?    ^16.?    '4123  .=» 

It  is  plain,  that  circulates  may  be  added,  subtracted,  multiplied,  and 
divided,  by  first  reducing  them  to  their  equivalent  vulgar  fractions. 


IT  75. 


REDUCTIOIf   OP   DECIMAL   FRACTIONS. 


145 


REDUCTION  OF  DECIMAL  FRACTIONS. 

IT  76-  Fractions,  we  have  seen,  (IT  63,)  like  integers^  are 
reduced  from  low  to  higher  denominations  by  division^  and 
from  high  to  lower  denominations  by  multiplication. 


To  reduce  a  compound  num- 
ber to  a  decimal  of  the  highest 
denomination, 

1.  Reduce  7  s.  6  d,  to  the 
decimal  of  a  pound. 

6  d.  reduced  to  the  decimal 
of  a  shilling,  that  is,  divided 
by  12,  is  '5  s.,  which  annexed 
to  the  7  s.  making  7'5  s.,  and 
divided  by  20,  is  '375  £,  the 
Ans. 

The  process  may  he  pre- 
sented in  form  of  arw/e,  thus : — 
Divide  the  lowest  denomina- 
tion given,  annexing  to  it  one 
or  more  ciphers,  as  may  be 
necessary,  by  that  number 
^^ilich  it  takes  of  the  same  to 
make  one  of  the  next  higher 
denomination,  and  annex  the 
quotient,  as  a  decimal  to  that 
higher  denomination;  so-con- 
tinue  to  do,  until  the  whole 
shall  be  reduced  to  the  deci- 
mal required. 

EXAMPLES  FOR  PRACTICE. 

3.  Reduce  1  oz.  10  pwt  to 
l)ie  fraction  of  a  poun^d. 

OPERATION. 
20)10^0     pwt. 

12)^5     oz. 

425  lb,  Ajis. 
N 


To  reduce  the  decimal  of  a 
higher  denomination  to  integers 
of  lower  denominations, 

2.  Reduce  '375  £> ,  to  in- 
tegers of  lower  denominations. 

^375  £ .  reduced  to  shillings, 
that  is,  multiplied  by  20,  is 
7^50  s. ;  then  the  fractimial 
part,  *50  s.,  reduced  to  pence, 
that  is,  multiplied  by  12,  is 
6  d.  Ans,  7  s.  6  d. 

That  is, — Multiply  the  given 
decimal  by  that  number  which 
it  takes  of  the  next  lower  de- 
nomination to  make  one  of  this 
higher,  and  from  the  right 
hand  of  the  product  point  off 
as  many  figures  for  decimals 
as  there  are  figures  in  the 
given  decimal,  and  so  con- 
tinue to  do  through  all  the  de- 
nominations; the  several  num* 
bers  at  the  left  hand  of  the 
decimal  points  will  be  the 
value  of  the  fraction  in  the 
proper  denominations. 

EXAMPLES  FOR  PRACTICE. 

4.  Reduce  425  lbs.  Troy  to 
integers  of  lower  denomina- 
tions. 

OPERATION. 
Ih.  n25 
_JL2 
oz.  1*500 

20 

pwt.  lO'OOO. 


Ans,  loz.lOpwt. 


REDUCTION    OP    DECIMAI4    FRACTIONS. 


Tr7e. 


6.  What  is    the   value   of 
*2325  of  a  ton? 


^8. 
hhd. 


What  is  the  value  of  ^72 
of  heer  ? 


t    10.  What  is  the  value  of 
'375  of  a  yard  ? 

k12.  Wliat  is  the  value  of 
'713  of  a  day? 

14.  What  is  the   value  of 
'78125  of  a  guinea  ? 

|16.  AVhat  is  the  value   of 
'15334821  of  a  ton? 


146 

; 

V  5.  Reduce  4  cwt.  2|  qrs.  to 
the  decimal  of  a  ton. 

,mte,     2{  —  2'6. 
.J^^7.  Reduce  38  gals.  3'52  qts. 
of  beer,  to  the  decimal  of  a 
hhd. 

ff    9.  Reduce  1  qr.  2  n.  to  the 
decimal  of  a  yard. 

11.  Reduce  17  h.  6  m.  43 
sec.  to  the  decimal  of  a  day. 

y  13.  Reduce  21  s.  10^  d.  to 
the  decimal  of  a  guinea. 

15.  Reduce  3  cwt.  0  qr.  7 
lbs.  8  oz.  to  the  decimal  of  a 
ton. 

Let  the  pupil  be  required  to  reverse  and  prove  the  follow- 
iufir  examples : 

yt,  R,educe  4  rods  to  the  decimal  of  an  acre, 

18.  What  is  the  value  of  '7  of  a  lb.  of  silver  ? 

19.  Reduce  18  hours.  15  m.  50'4  sec.  to  the  decimal  of  a 
day. 

20.  W^hat  is  the  value  of  '67  of  a  league  ? 

21.  Reduce  10  s.  9J  d.  to  the  fraction  of  a  pound. 

IT  fi^S.  There  is  a  method  of  reducing  shillings,  pence 
and  farthings  to  the  decimal  of  a  pound,  by  inspection^  more 
simple  and  concise  than  the  foregoing.  The  reasoning  in 
relation  to  it  is  as  follows  : 

yV  of  20  s.  is  2  s. ;  therefore  every  2  s.  is  yV?  or  4  i8. 
Every  shilling  is  ^^  =  y§^,  or  '05  £ .  Pence  are  readily- 
reduced  to  farthings.  Every  farthing  is  ^^  M .  Had  it  so 
happened,  that  1000  farthings,  instead  of  960,  had  made  a 
pound,  then  every  farthing  would  have  been  y^Vry?  or  '001  £, . 
But  960  increased  by  ^  part  of  itself  is  1000;  conse- 
quently, 24  farthings  are  exactly  to  ^tt)  o^  ^^25  £> .,  and  48 
farthings  are  exactly  yf  Htj-,  or  '050  £ .  Wherefore,  if  the 
number  of  farthings,  iu  the  given  pence  and  farthings,  be 
more  than  12,  ^  part  will  be  more  than  ^;  therefore  add  1 
to  them :  if  they  be  more  than  36,  ^  part  will  be  more  than 
1^;  therefore  add  2  to  them:  then  call  them  so  many* 
thousandths,  and  the  result  will  be  correct  within  less  than 
i  of  TtrW  of  a  pound.     Thus,  17  s.  6f  d.  is  reduced  to  tha 


1r77.  REDUCTION    OF    DECIMAL  FRACTIONS.  147 

decimal  of  a  pound  as  follows  :  16  s.  =  *8  i6 .  and  1  s.  = 
'05  £> .  Then,  5 J  d.  =  23  farthings,  which^  increased  by 
1,  (the  number  being  more  than  12,  but  not  exceeding  36,)  is 
<024  £ .,  and  the  whole  is  S74  £ .  the  Am. 

Wherefore,  to  reduce  shillings^  pence  and  farthings  to  the 
decimal  of  a  pound  by  inspection^ — Call  every  two  shillings  one 
tenth  of  a  pound  ;  every  odd  shilling^  five  hundredths  ;  and  the 
number  of  farthings^  in  the  given  pence  and  farthings^  so  ma?nf 
thousandths^  adding  one^  if  the  number  be  more  than  twelve  and 
not  exceeding  thirty-six^  and  tivo,  if  the  number  be  more  than 
thirty-six, 

IT  77.  Reasoning  as  above,  the  result,  or  the  three  first 
figures  in  any  decimal  of  a  pound,  may  readily  be  reduced 
back  to  shillings,  pence  and  farthings,  by  inspection.  Double 
the  first  figure,  or  tenths^  for  shillings,  and,  if  the  second 
figure,  or  hundredths,  be  five,  or  more  than  five,  reckon  ano- 
ther shilling ;  then,  after  the  five  is  deducted,  call  the  figures 
in  the  second  and  tlJid  place  so  many  farthingG,  abating 
one  when  they  are  above  twelve,  and  two  when  above  thir- 
ty-six, and  the  result  will  be  the  answer,  sufficiently  exact 
for  all  practical  purposes.  Thus,  to  find  the  value  of  ^876  £ . 
by  inspection  : — 

*8      tenths  of  a  pound  -         -         -         =z  16  shillings. 

*05    hundredths  of  a  pound  -         -         zz:     1  shilling. 

*026  thousandths,  abating  1,  =:  25  farthings  zir     0  s.     G^d, 

'876  of  a  pound  -        -         -        -         z=  17  s.     6^  d. 

Am, 
EXAMPLES   FOR   PRACTICE. 

1.  Find,  by  inspection,  the  decimal  expressions  of  9  s.  7  d., 
and  12  s.  Of  d.  Am,  '479^2.,  and  '603^2. 

2.  Find,  by  inspection,  the  value  of  ^523^3.,  and  '691^2. 

Ans.  10  s.  5^  d.,  and  13  s.  10^  d. 

3.  Reduce  to  decimals,  by  inspection,  the  following  sums, 
and  find  their  amonnt,  viz. :  15  s.  3  d. ;  8  s.  11^  d. ;  10  s. 
6^  d. ;  1  s.  8^  d.  ;  J  d.,  and  2^  d.  Atnouht,  £  1'833. 

4.  Find  the  value  of  '47  £, 

•  Note,  When  the  decimal  has  but  two  figures,  after  taking 
out  the  shillings,  the  remainder,  to  be  reduced  to  thousandths^ 
will  require  a  cipher  to  be  annexed  to  the  right  hand,  or 
supposed  to  be  so.  Ans,  9  s.  4|  i?. 


148  SUPPLEMENT  TO  DECIMAL  FRACTIONS.  IF  77. 

5.  Value  the  following  decimals,  by  inspection,  and  find 
their  amount,  viz.;  *'785i^.;  '357  £.;  '916  i^.;  '74  JS.; 
'6£.;  '26£.;  '09  JB.;  and '008  iS.     ^?i6.  3JS.  12s.  11  d. 


SUFS»Z.EM£NT   TO  DECZMAX.    FHACTZONS. 

QUESTIONS. 

1.  What  are  decimal  fractions  ?  2.  Whence  is  the  term 
derived  ?  3.  How  do  decimal  differ  from  coriimon  frac- 
tions ?  4.  How  are  decimal  fractions  written  ?  6.  How 
can  the  proper  denominator  to  a  decimal  fraction  be  known, 
if  it  be  not  expressed  ?  6.  Hew  is  the  value  of  ©very  figure 
determined?     7.  What  does  the  first  figure  on  the  right 

hand  of  the  decimal  point  signify  ?     the  second  figure  ? 

< third  figure  ?    fourth  figure  ?    8.  How  do  ciphers^ 

placed  at  the  right  hand  of  decimals,   affect  their  value  ? 

9.  Placed  at  the  left  hand,  how  do  they  affect  their  value  ? 

10.  How  are  decimJs  read?  11.  How  are  decimal  frac- 
tions, having  different  denominators,  reduced  to  a  common 
denominator?  12.  What  is  a  mixed  number?  13.  How 
may  any  iclwle  number  be  reduced  to  decimal  parts  ?  14. 
How  can  any  mixed  number  be  read  together,  and  the 
whole  expressed  in  the  form  of  a  common  fraction  ?  15. 
What  is  observed  respecting  the  denominations  in  federal 
money  ?  16.  What  is  the  rule  for  addition  and  subtraction 
of  decimals,  particularly  as  respects   placing  the   decimal 

point  in  the  results  ?     multiplication  ?     division  ? 

17.  How  is  a  common  or  vulgar  fraction  reduced  to  a  deci- 
mal ?  18.  What  is  the  rule  for  reducing  a  compound  num- 
ber to  a  decimal  of  the  highest  denomination  contained  in 
it  ?  19.  What  is  the  rule  for  finding  the  value  of  any  given 
decimal  of  a  higher  denomination  in  terms  of  a  lower  f 
20.  What  is  the  rule  for  reducing  shillings,  pence  and  far- 
things to  the  decimal  of  a  pound,  by  inspectmi  ?  21.  What 
^i^the  reasoning  in  relation  to  this  rule  ?  22.  How  may  the 
three  first  figures  of  any  decimal  of  a  pound  be  reduced  to« 
shillings,  pence  and  farthings,  by  insjiectwn  1 


M  UK.         SUPPLEMENT   TO   BECIMAt.   FRACTIONS.  149 

EXERCISES. 

1.  A  merchant  had  several  remnants  of  cloth,  measuring 
as  follows,  viz.  : 


7 1  yds. 
6f  

1  i  

9f  

8i   

3jV 


How  many  yapds  in  the  whole,  and  what  would 
the  whole  come  to  at  $  3'67  per  yard  ? 


Note,  Reduce  the  common  fractions  to  deci- 
mals. Do  the  same  where var  they  occur  in  the 
examples  which  follow. 

Ans,  36^475  yards.      $  133^863  +,  cost. 

J^.  2.  From  a  piece  of  cloth,  containing  36|  yds,,  a  merchant 
sold,  at  one  time,  7^^^  yds.,  and,  at  another  time,  12|  yds. ; 
how  much  of  the  cloth  had  he  left  ?  Ans,  IG'7  yds. 

3.  A  farmer  bought  7  yards  of  broadcloth  for  Sy^  i3 .,  a 
barrel  of  flour  for  2y\  ^ .,  a  cask  of  Hme  for  1|  £ .,  and  7  lbs. 
of  rice  for  ^  jS.  ;  he  paid  1  ton  of  hay  at  3^^^  £,^  I  cow 
at6f  £.j  and  the  balance  in  pork  at  -^^  £,  per  lb.;  how 
many  were  the  pounds  of  pork  ? 

Note,  In  reduci»:ig  the  common  fractions  in  this  example, 
it  will  be  sufficiently  exact  if  the  decimal  be  extended  to 
three  places.  Ans.  l()8f  lb. 

^^  4.  At  12^  cents  per  lb.,  what  will  37|-  lbs.  of  butter  cost? 

-4715.   $4'718}. 

5.  At  $17*37  per  ton  for  hay,  what  will  llf  tons  cost? 

Ans.  $201'92f. 

6.  TTie  above  example  reversed.  At  $  201 '92|  for  1 1 1  tons 
of  hay,  what  is  that  per  ton  ?  Ans.  $  17*37. 
I  7.  If  *45  of  a  ton  of  hay  cost  $  9,  what  is  that  per  ton  ? 
Co7isult  'W  65.  Ans.  $  20. 

8.  At  '4  of  a  dollar  a  gallon,  what  will  *25  of  a  gallon 
of  molasses  cost  ?  Ans.  $*1. 

9.  At  $  9  per  cwt.,  what  will  7  cwt.  3  qrs.  IG  lbs.  of  sugar 
cost? 

Note.  Reduce  the  3  qrs.  16  lbs.  to  the  decimal  of  a  cwt, 
extending  the  decimal  in  this,  and  the  examples  which  fol- 
low, to  four  places.  Ans.  71'035-(-, 

10.  At  $69*875  for  5  cwt.  1  qr.  14  lbs.  of  raisins,  what  is 
that  per  cwt.  ?  Ans.  $  13. 

11.  What  will  2300  lbs.  of  hay  come  to  at  7  mills  per  )b.  ? 

Ans,  $  itinO. 

12.  What  will  765^  lbs.  of  coffee  come  to,  at  18  cents  per 
lb.?  Ans,  $137*79 


150 


SUPPLEMENT    TO   DECIMAL    FllACTIOlfSv         f  77/ 


13.  What  will  12  gals.  3  qts.  1  pt  of  gin  cost,  at  28  cents 
per  quart? 

Note.     Reduce  the  whole  quantity  to  quarts  and  the  deci- 
mal of  a  quart.  Am.  $  J,4'42. 
114.  Bought  16  yds.  2  qrs.  3  na,  of  broadcloth  for  $  100*125 ; 
what  was  tliat  per  yard  ?  Am.  $  6. 

15.  At  $  1*92  per  bushel,  how  much  wheat  may  be 
bought  for  $  '72  ?  Ans.  1  peck  4  quarts- 

fl6.  At  $  92'72  per  ton,  how  much  iron  may  be  pur- 
chased for  .$60'268r  Ans.  13  cwt. 

17.  Bought  a  load  of  hay  for  $947,  paying  at  the  rate 
of  $  16  per  ton  j  what  was  the  weight  of  the  hay  ? 

A71S.  11  cwt.  1  qr.23lbs. 

/18.  At  $302*4  per  tun,  what  will  1  hhd.  15  gals.  3  qts. 

of  wine  cost  ?  Am.  $  94*50. 

19.  The  above  reversed.  At  $94*50  for  1  hhd.  15  gals. 
3  qts.  of  wine,  what  is  that  per  tun  ?  Am.  $  302*4. 

Note.  The  following  examples  reciprocally  prove  each 
other,  excepting  when  there  are  some  fractional  losses,  as  ex- 
plained above,  and  even  then  the  results  will  be  sufficiently 
exact  for  all  practical  purposes.  If,  however,  greater  exact- 
ness be  required,  the  decimals  must  be  extended  to  a  greater 
number  of  places. 


20.  At  $1*80  for  3^  qts.  of 
wine,  what  is  that  per  gal.  ? 

22.  If  f  of  a  ton  of  pot- 
ashes cost  $60*45,  what  is 
that  per  ton  ? 


21.  At  $2*215  per  gal., 
what  cost  3-j[-  qts.  ? 

23.  At  $96*72  per  ton  for 
pot-ashes,  what  will  f  of  a  ton 
cost  ? 


24-  If*Sofayard 
of  cloth  cost  $2, 
what  is  that  per 
>  ard  > 

27.  If  14  cwt.  of 
pot-ashes  cost  19  JS . 
5  8.,  what  is  that 
per  ton  ? 


25.  If  a  yard  of 
cloth  cost  $  2*5, 
what  will  *8  of  a 
yard  cost  ? 

28.  If  a  ton  of 
pot-ashes  cost  27 iB . 
10  s.,  what  will  14 
cwt.  cost  ? 


26.  At  $2*5  per 
yard,  how  much 
cloth  may  be  pur- 
chased for  $  2  ? 

29.  At 27 ie.  10s. 
a  ton  for  pot-ashes, 
what  quantity  may 
be  bought  for  19  iS. 
5  s.? 


Note,    After  the  same  manner  let  the  pupil  reverse  and 
prove  the  following  examples  : 


ir77j78.  REDUCTION    OF   CURRENCIES.  151 

30.  At  $  18^50  per  ton,  how  much  hay  may  be  bought 
for  $  12^025  ? 

31.  What  will  3  qrs.  2  na.  of  broadcloth  cost,  at  $6  pet 
yard  ? 

^'32.  At  $  2240  for  transportation  of  65  cwt.  46  miles,  what 
is  that  per  ton  ? 

33.  Bought  a  silver  cup,  weighing  9  6z.  4  pwt.  16  grs.  foi* 
3  iS .  2  s.  3  d.  3|  q. ;  what  was  that  per  ounce  ? 

34.  Bought  9  chests  of  tea,  each  weighing  3  cwt.  2  qrs.  21 
lbs.  at  4  dB .  9  s.  per  cwt. ;    what  came  they  to  ? 

35.  If  5  acres  1  rood  produce  26  quarters  2  bushels  of 
wheat,  how  many  acres  will  be  required  to  produce  47 
quarters  4  bushels  ?  A  quarter  is  8  bushels. 

Note.  The  above  example  will  require  two  operations, 
for  which  consult  H  65,  ex.  1. 

//  36.  A  lady  purchased  a  gold  ring,  giving  at  the  rate  of 
*$  20  per  ounce ;  she  paid  for  the  ring  $1^25;  how  much 
did  it  weigh  ? 


RliDUGTZO^I  OF   CUHRENCIBS. 

IT  78-  Previous  to  the  act  of  Congress  in  1786  establish-' 
ing  federal  money,  all  calculations  in  money,  throughout  the 
United  States,  were  made  in  pounds,  shillings,  pence  and 
farthings,  the  same  as  in  PJngland.  But  these  denominations, 
although  the  same  in  name,  were  different  in  value  in  dii^ 
ferent  countries. 

Thus,  1  dollar  is  reckoned  in 

England,  4  s.  6  d.,  called  English^  or  sterling  money. 


Nova'^Sc^^fa       \  ^  ^'  ^^^^^  Canada  currency. 

^  6  s.,  called  New  England  curreacf  ^ 


The  New  Eng-^ 

land  States, 
Virginia, 
Kentucky,  and 
Tennessee, 
New  York,         ) 

Ohio,  and  /8s.,  called  New  Yhrk  currency. 

N.  Carolina,      ) 


153  REDUCTION    OF   CURRENCIES.  IT  78 

1  dollar  is  reckoned  in 


7  s.  6  d.,  called  Pennsylvania  currency 


New  Jersey, 
Pennsylvania, 
Delaware,  and 
Maryland, 

GeorS"''  ^""^  1  ^  '•  ^  ^''  '"'■"^'^  ^"^^'"  currency. 
1.  Reduce  6£,  11  s.  6^-d.  to  federal  money. 

Note,  To  reduce  pounds,  shillings,  pence  and  farthings, 
in  either  of  the  above-named  currencies,  to  federal  money, — 
First,  reduce  the  shillings,  pence  and  farthings  (if  any  be 
contained  in  the  given  sum)  to  the  decimal  of  a  pound  by  in- 
spection^ as  already  taught,  IF  76. 

6£.  lis.  6id.  —  £6'576. 

English  money. — Now,  supposing  the  above  sum  to  be 
bJnglish  money, — 1£,  is  20  s.  =:  240  pence,  in  all  the  above 
currencies.  1  dollar,  in  English  money,  is  reckoned  4  s.  6  d. 
=:  54  pence,  that  is,  ^^/^j  =  ^u  of  -t  pound.  Now,  as  many 
times  as  -^tj^  the  fraction  which  1  dollar  is  of  1  pound,  Eng- 
lish money,  is  contained  in  J2  6 '576,  so  many  dollars,  it  is 
evident,  there  must  be ;  that  is, —  7b  reduce  English  to  federal 
money ^ — Divide  the  given  sum  by  £^^  the  quotient  will  be 
federal  money. 

je  6  W6  English  money.  ^ote.     It  will  be 

40  recollected,  to    di- 

vide  by  a  fraction, 

9)  263'040  we  multiply  by  the 

l^^f  federal  money,  Answer.         tidf  ihfp'roduc't 

by  the  numerator. 

Canada  currency. — Supposing  the  above  sum  to  be  Cana- 
da currency, — 1  dollar,  in  this  currency,  is  5  s.  :=  60  pence, 
that  is,  -^-^^  z=  ;|-  of  1  pound.  Therefore, —  To  reduce  Canor 
da  currency  to  federal  money ^ — Divide  the  given  sum  by  J,  and 
the  quotient  will  be  federal  money  \  or,  which  is  the  same 
thing, — Multiply  the  given  sum  by  4. 

£6*576  Canada  currency. 
4 


$  27'304  federal  money,  Answer. 


V  Tf8.  REDUCTION    OF    CURRENCIES.  16S 

New  England  currency. — 1  dollar,  in  this  currency,  is 
6  s.  n:  72  pence,  that  is,  -^^jj  =:  -f^j  or  '3  of  a  pound.  There- 
fore,— To  reduce  New  England  currency  to  federal  money j — Di- 
vide the  given  sum  by  '3. 

'3)  £ .  6*576  New  England  currency. 

$21*92  federal  money,  Answer. 
New  York  currency. — 1  dollar,  in  this  currency,  is  8  s.  = 
96  pence,  that  is,  -r?^^  =  •^,  or  '4  of  a  pound.     Therefore, 
— To  reduce  New  York  currency  to  federal  money y — Divide  the 
given  sum  by  '4. 

*4)  £ .  6*576  New  York  currency. 

$  16*44  federal  money.  Answer, 
Pennsylvania  crRRENcv. — 1  dollar, in  this  currency, is  75v 
e  d.  =  90  pence,  that  is,  ^^^  ■=.  f  of  a  pound.  Therefore, — 
To  reduce  Pennsylvania  currency  to  federal  money ^ — Divide  by 
§,  that  is,  multiply  the  given  sum  by  8,  and  divide  the  pro- 
duct by  3. 

£ .  6*576  Pennsylvania  currency. 
8 


3)52*608 

$  17*536  federal  money,  Answer, 
Georgia  currency. — 1  dollar,  Georgia  currency,  is  4  »* 
8  d.  :=  56  pence,  that  is,  ^y^y  z=z  ^^  of  a  pound.  Therefore, — 
To  reduce  Georgia  currency  to  federal  money, — Divide  by  -^j 
that  is,  multiply  the  given  sum  by  30,  and  divide  the  pro- 
duct by  7. 

£ .  6*576  Georgia  currency. 
30 


7)197*280 


$28*182f-  federal  money.  Answer, 

From  the  forec^oing  examples,  we  derive  the  following 
general  Rule  : — To  reduce  English  money,  and  the  currencies 
of  Canada  and  the  several  States,  to  federal  money, — First,  re- 
duce the  shillings,  &c.,  if  any  in  the  given  sum,  to  the  deci- 
mal of  a  pound;  this  being  done,  divide  the  given  sum  by 
such  fractional  part  as  1  dollar,  in  the  given  currency,  it 
a  fractional  part  of  1  pound. 


152  REDUCTION    OF   CURRENCIES.  IT  78 

1  dollar  is  reckoned  in 
New  Jersey,      ^ 

Kwi'erand    f  ^  '*  ^  ^'^  ^^"^^  Pennsylvania  currency 

Maryland,  J 

S.  Carolina  and  >  .      ^  ,        n  j  /-r 

Georo-ia,  j  4  s.  8  d.,  called  Georgia  currency. 

1.  Reduce  6iB.  11  s.  6;^d.  to  federal  money. 

Note,  To  reduce  pounds,  shillings,  pence  and  farthings, 
in  either  of  the  above-named  currencies,  to  federal  money, — 
First,  reduce  the  shillings,  pence  and  farthings  (if  any  be 
contained  in  the  given  sum)  to  the  deciinal  of  a  pound  by  i»- 
spection^  as  already  taught,  U  76. 

6£,  lis.  6^d.  ~  £6W6. 
English  money. — Now,  supposing  the  above  sum  to  be 
bJnglish  money, — 1^ .  is  20  s.  =z  240  pence,  in  all  the  above 
currencies.  1  dollar,  in  English  money,  is  reckoned  4  s.  6  d. 
=z  54  pence,  that  is,  ^^/tj  =  /^y  of  1  pound.  Now,  as  many 
times  as  ;j^75-,  the  fraction  which  1  dollar  is  of  1  pound,  Eng- 
lish money,  is  contained  in  i26'576,  so  many  dollars,  it  is 
evident,  there  must  be ;  that  is, —  To  reduce  English  to  federal 
money ^ — Divide  the  given  sum  by  ^^^  the  quotient  will  be 
federal  money. 

je  6  W6  English  money.  ^ote.     It  will  be 

40  recollected,  to   di- 

vide  by  a  fraction, 

9) 263^040  we  multiply  by  the 

"io^f  federal  money,  ^n.s«.^.  denominator,     and 

**  •"  divide  the  product 

by  the  numerator. 

Canada  currency. — Supposing  the  above  sum  to  be  Cana- 
da currency, — 1  dollar,  in  this  currency,  is  5  s.  m  60  pence, 
that  is,  -^^  =  ;|-  of  1  pound.  Therefore, —  To  reduce  Canor 
da  currency  to  federal  money ^ — Divide  the  given  sum  by  ^,  and 
the  quotient  will  be  federal  money ;  or,  which  is  the  same 
thing, — Multiply  the  given  sum  by  4. 

i£6'576  Canada  currency. 
4 


$  27'304  federal  money,  Answer. 


U   T^8.  REDUCTION    OF    CURRENCIES.  15^ 

New  England  currency. — 1  dollar,  in  this  currency,  is 
6  s.  r=  72  pence,  that  is,  -^-^jj  =.  -^^^  or  '3  of  a  pound.  There- 
fore,— To  reduce  New  England  currency  to  federal  money j — Di^ 
vide  the  given  sum  by  *3. 

*3)  £ .  6*576  New  England  currency. 

$21 '92  federal  money.  Answer. 
New  York  currency. — 1  dollar,  in  this  currency,  is  8  s.  = 
96  pence,  that  is,  ^^^  =z  -^,  or  '4  of  a  pound.     Therefore, 
—To  reduce  New  York  currency  to  federal  money ^ — Divide  the 
given  sum  by  '4. 

'4)  £ .  6 '576  New  York  currency. 

$  16'44  federal  money.  Answer. 
Pennsylvania  currency. — 1  dollar,  in  this  currency,  is  75. 
6  d.  z=  90  pence,  that  is,  ^\%  i=  f  of  a  pound.  Therefore, — 
To  reduce  Pennsylvania  currency  to  federal  money, — Divide  by 
§,  that  is,  multiply  the  given  sum  by  8,  and  divide  the  pro- 
duct by  3. 

£ .  6 '576  Pennsylvania  currency. 
8 


3)52'608 


$  17'536  federal  money,  Answer. 
Georgia  currency. — 1  dollar,  Georgia  currency,  is  4  »* 
8  d.  1=  56  pence,  that  is,  ^\^j  z=z  -^^  of  a  pound.  Therefore, — 
To  reduce  Georgia  currency  to  federal  money, — Divide  by  -^j 
that  is,  multiply  the  given  sum  by  30,  and  divide  the  pro- 
duct by  7. 

£ .  6'576  Georgia  currency. 
30 


7)197'280 


$28'182f-  federal  money,  Answer. 

From  the  foresjoing  examples,  we  derive  the  following 
general  Rule  : — To  reduce  English  money,  and  the  currencies 
of  Canada  and  the  several  States,  to  federal  money, — First,  re- 
duce the  shillings,  &c.,  if  any  in  the  given  sum,  to  the  deci- 
mal of  a  pound;  this  being  done,  divide  the  given  sum  by 
such  fractional  part  as  1  dollar,  in  the  given  currency,  Jg 
a  fractional  part  of  1  pound. 


166  INTEREST.  IT  80, 81. 


Rates  at  which  the  following  foreign  coins  are  estimated  at  the 
Custom  Houses  of  the  United  States, 

Livre  of  France,      ----------  $    qg^ 

Franc         do.  $    4Sf, 

Silver  Rouble  of  Russia,    --------  ^    '75. 

Florin  or  Guilder  of  the  United  Netherlands,      -  $    *40. 

Mark  Banco  of  Hamburg,       «------  ^    ^33^ 

Real  of  Plate  of  Spain,      --------  ^    40. 

Real  of  Vellon  of  do. $    <05. 

Milrea  of  Portugal,        ---------  $1*24. 

Tale  of  China, $1^48. 

Pagoda  of  India,      ----------  ^  1'84. 

Rupee  of  Bengal,     ----------  ^    '50, 

2.  Reduce  8764  livres  to  federal  money. 

3.  Reduce  10,000  francs  to  federal  money. 

4.  Reduce  250,000  florins  to  federal  money. 
6.  In  $  1000,  how  many  francs  ? 


IlffTERBST. 

T[  81.  In-terest  is  an  allowance  made  by  a  debtor  to  a 
creditor  for  the  use  of  money.  It  is  computed  at  a  certain 
number  of  dollars  for  the  use  of  each  hundred  dollars,  or  so 
many  pounds  for  each  hundred  pounds,  &c.  one  year,  and 
in  the  same  proportion  for  a  greater  or  less  sum,  or  for  a 
longer  or  shorter  time. 

The  number  of  dollars  so  paid  for  the  u.^e  of  a  hundred 
dollars,  one  year,  is  called  the  rate  per  cent,  or  per  centum  ; 
the  words  per  cent,  or  per  centum  signifying  by  the  hundred. 

The  highest  rate  allowed  by  law  in  the  New  England 
States,  is  6  per  certt.*  that  is,  6  dollars  for  a  100  dollars,  6 
cents  for  a  100  cents,  6  pounds  for  a  100,  &c. ;  in  other 
words,  Y§^  of  the  sum  lent  or  due  is  paid  for  the  use  of  it  one 
year.  This  is  called  legal  interest^  and  will  here  be  under- 
utood  when  no  other  rate  is  mentioned. 


♦  In  the  State  of  New  York;  7  per  cent  is  the  le^^al  interest  j  in  Enjlend  th« 
twjiu  iirteresi  is  5  per  cent. 


IT  8 1 .  INTEREST.  ]  57 

Let  us  suppose  the  sum  lent,  or  due^  to  be  $  1.  The 
100th  part  of  $  1,  or  ^^-^  of  a  dollar,  is  1  cent,  and  yf  ^y  of  a 
dollar,  the  legal  interest,  is  6  cents,  which,  written  as  a  de- 
cimal fraction,  is  expressed  thus,     ------     <06. 

So  of  any  other  rate  per  cent. 

1  per  cent.,  expressed  as  a  common  fraction,  is 
y^;  decimally,      -----------     *oi. 

-j  per  cent,  is  a  half  of  1  per  cent.,  that  is,       -    -     ^005. 

^  per  cent,  is  a  fourth  of  1  per  cent.,  that  is,  -    -     '0025. 

J  per  cent,  is  3  times  J  per  cent.,  that  is,  -    -     -     '0075. 

Note,  The  rate  per  cent,  is  a  decimal  carried  to  two 
placeSy  that  is,  to  hundredths ;  all  decimal  expressions  lower 
than  hundredths  are  parts  of  1  pei*  cent.  |  per  cent.,  for  in- 
stance, is  *625  of  1  per  cent.,  that  is,  '00625. 


Ans.  '025. 


Write  2J  per  cent,  as  a  decimal  fraction. 

2  per  cent,  is  '02,  and  ^  per  cent,  is  '005.  Ans.  *U25. 

Write  4  per  cent,  as  a  decimal  fraction.    4J  per 

cent.     4|-  per  cent.    5  per  cent.    7^  per 

cent.    8  per  cent.    8|  per  cent.     9  per 

cent.    9 J  per  cent.    10  per  cent.   (10  per  cent. 

is  -x^;  decimally, '10.)    10^  per  cent.    11  per 

cent.    12^  per  cent. 15  per  cent. 

1.  If  the  interest  on  $  1,  for  1  year,  be  6  cents,  what  will 
be  the  interest  on  $  17  for  the  same  time  ? 

It  will  be  17  times  6  cents,  or  6  times  17,  which  is  the 
same  thing : — 

$17 
'06 

1'02  Answer  ;  that  is,  1  dollar  and  2  cents. 

To  fiad  the  interest  on  any  sum  for  1  year,  it  is  evidei|t 
we  need  only  to  multiply  it  by  the  rate  per  cent,  written  as  a 
decimal  fraction.  The  product,  observing  to  place  the  point 
as  directed  in  multiplication  of  decimal  fractions,  will  be  the 
interest  required. 

NotfiiA  Principal  is  the  money  diiCj  for  which  interest  i« 
paid.     Amount  is  the  principal  and  interest  added  together. 

O 


158  INTEREST.  1181,82 

2.  What  will  be  the  interest  of  $  3245,  1  year,  at  4^  per 
cent.  ? 

$'32'16  principal  rj.^^^^  ^^^^^  five  de- 

__^045  rate  per  cent.  ^-^^^^  pl^^^g  jn  the  mul- 

jg()Y5  tiplicand  and  multiplier, 

22860  ^^^'^    iigiires     must    be 

pointed  olf  for  decimals 


Afis.    $  1 '446 75  from  the  product,  which 

gives  the  answer, — 1 
dollar,  44  cents,  G  mills,  and  -^^^^  of  a  mill.  Parts  of  a  mill  are 
not  generally  regarded;  hence,  $  1^446  is  sufficiently  exact 
for  the  answer.  ,; 

3.  What  will  be  the  interest  of  $  11^04  for  1  year,  at  3 

per  cent.  ?  .  at  5^  per  cent.  ?  at  6  per  cent.  ?  

at  7}  per  cent.  ?  at  8^- per  cent.  ?  at  9£  per  cent,  ? 

at  10  per  cent.  ?    at   10^  per  cent.  ?  at  11 

percent.?  at  llf  per  cent.?    at   12  per  cent.: 

at  12J  per  cent.  ? 

f  4.  A  tax  on  a  certain  town  is  $  1627^18,  on  which  the 
collector  is  to  receive  2^  per  cent,  for  collecting;  what  will 
he  receive  for  collecting  the  whole  tax  at  that  rate  ? 

Alls.    $40^679, 

JVoie.  In  the  same  way  are  calculated  commission,  irv 
surance,  buying  and  selling  stocks,  loss  and  gain,  or  any 
tfii])g  else  rated  at  so  much  per  cent,  without  respect  to  time. 

{  6.  What  must  a  man,  paying  $  0'374  on  a  dollar,  pay  on 
a  debt  of  $  132'25  ?  "  Ans.    $  49'593. 

6.  A  merchant,  having  purchased  goods  to  the  amount  ol 
$  580,  sold  them  so  as  to  gain  12^  per  cent.,  that  is,  12^ 
cents  on  each  100  cents,  and  in  the  same  proportion  for  a 
greater  or  less  sum  ;  what  was  his  whole  gain,  and  what  was 
the  whole  amount  for  which  he  sold  the  goods  ? 

Ans.  His  whole  gain  was  $  72^50  ;  whole  amount 
^,  652^50. 

7.  A  merchant  bought  a  quantity  of  goods  for  $  763*37^ ; 
liow  much  must  he  sell  them  for  to  gain  15  per  cent. .? 

Ans.    $  877^881. 


ir  82.  Commission  is  an  allowance  of  so  much  per  cent- 
to. a  person  called  a  correspondenty factor,  or  broker,  for  as- 
Bksting  merchants  and  others  in  purchasing  and  selling  goods. 


IT  82.  INTERESl.  159 

^  8.  My  correspondent  sends  me  word  that  he  has  pur- 
chased goods  to  the  vahie  of  $  1286,  on  my  account;  what 
will  his  commission  come  to  at  2^  per  cent.  ?    Ans.  $32'15. 

^9.  What  must  I  allow  my  correspondent  for  selling  goods 
to  the  amount  of  $  2317'46,  at  a  commission  of  3^  per  cent.  ? 

Ans.    $75^317. 


Insurance  is  an  exemption  from  hazard,  obtained  by  tlie 
payment  of  a  certain  sum,  which  is  generally  so  much  per 
cent,  on  the  estimated  value  of  the  property  insured. 

Premium  is  the  sum  paid  by  the  insured  for  the  insurance. 

Policy  is  the  name  given  to  the  instrument  or  writing, 
by  which  the  contract  of  indemnity  is  effected  between  the 
insurer  and  insured. 

10.  What  will  be  tha  premium  for  insuring  a  ship  and 

cargo  from  Boston  to  Amsterdam,  valued  at   $  37800,  at  4^ 

per  cent.  ?  c  Ans.    $  170f. 

'11.  What  will  be  the  annual  premium  for  insurance  on  a 

house  against  loss  from  fire,  valued  at  $  3500,  at  f  per  cent.  ? 

By  removing  the  separatrix  2  figures  towards  the  left,  it  is 
evident,  the  sum  itself  may  be  made  to  express  the  premium 
at  1  per  cent.,  of  which  the  given  rate  parts  may  be  taken ; 
thus,  1  per  cent,  on  $  3500  is  $  35^00,  and  f  of  $  35*00  is 
$  26*25,  Answer. 
/  12.  What  will  be  the  premium  for  insurance  on  a  ship  and 

cargo  valued  at  $25156*86,  at  ^  per  cent.  ?  at  f  per 

cent.  ?  ^at  f  per  cent.  ?  at  f  per  cent  ?  at  f 

per  cent.  ?       Ans.  At  |-  per  cent,  the  premium  is  $  157*23. 


Stock  is  a  general  name  for  the  capital  of  any  trading 
company  or  corporation,  or  of  a  fund  established  by  govern- 
ment. 

The  value  of  stock  is  variable.  When  100  dollars  of 
stock  sells  for  100  dollars  in  money ^  the  stock  is  said  to  be  at 
par,  which  is  a  Latin  word  signifying  equal ;  when  for  7nore^ 
it  is  said  to  be  aboMt  par ;  when  for  less^  it  is  said  to  be  6e- 
low  par. 

'  13.  What  is  the  value  of   $7564  of  stock,  at  112^  per 
cent. :  that  is,  when  1  dollar  of  stock  sells  for  1  dollar  12^ 


160  INTEREST.  ^^  !I  82, 83 

cents  in  tnoney^  which  is  12^  per  cent,  above  par,  or  12^  per 

ceat.  advance^  as  it  is  sometimes  called.         Am,   $  8509*50. 

^4.  What  is  the  value  of   $3700  of  bank  stock,  at  95^ 

per  cent.,  that  is,  4J  per  cent,  helow  par  ?     Am.    $  35a3^50. 

15.  What  is  the  value  of  $  120  of  stock,  at  92 J  per  cent.  ? 

at  86  J  per  cent.  ? at  67f  per  cent.  ?  at  104^ 

per  cent.  ?  at  lOS^  per  cent.  ?  at  115  per  cent  ? 

at  37^-  per  cent,  advance  ? 


/  Loss  AND  Gain.     16.  Bought  a  hogshead  of  molasses  for 
$  60  ;  for  how  much  must  I  sell  it  to  gain  20  per  cent.  ? 

Am,    $  72. 

^17.  Bought  broadcloth  at  $2'50per  yard;  but,  it  being 

damaged,  I  am  willing  to  sell  it  so  as  to  lose  12  per  cent ; 

how  much  will  it  be  per  yard  ?  Am,    $  2*20. 

'  18.  Bought  calico  at  20  cents  per  yard ;  how  must  I  sell  it 

to  gain  5  per  cent.  ?  10  per  cent.  ?  15  per  cent  ? 

to  lose  20  per  cent.  ?     Aits,  to  the  lasty  16  cents  per  yard. 

^  83.  We  have  seen  how  interest  is  cast  on  any  sum  of 
money,  when  the  time  is  one  year ;  but  it  is  frequently  ne- 
cessary to  cast  interest  for  months  and  days. 

Now,  the  interest  on  $  1  for  I  year,  at  6  per  cent.,  being 
*06,  is 

*01  cent  for  2  months, 
*005  mills  (or  ^  a  cent)  for  1  month  of  30  days,  (for  so  we 

reckon  a  month  in  casting  interest,)  and 
*001  mill  for  every  6  days ;  6  being  contained  5  times  in  30. 

Hence,  it  is  very  easy  to  find  by  insj^ection,  that  is,  to  cast 
in  the  mind,  the  interest  on  1  dollar,  at  6  per  cent  for  any 
given  iime.  The  cents^  it  is  evident,  will  be  equal  to  /ia(/*the 
greatest  even  number  of  the  months  ;  the  mills  will  be  5  for 
the  odd  month,  if  there  be  one,  and  1  for  every  time  6  is 
contained  in  the  given  number  of  the  days. 

Suppose  the  interest  of  $  1,  at  6  per  cent,  be  required  for 
9  months,  and  18  days.  The  greatest  even  number  of  the 
months  is  8  half  of  which  will  be  the  cents,  *04 ;  the  mills, 
reckoning  5  for  the  odd  month,  and  3  for  the  18  (3  times  6 
=  IS\  days,  will  be  *008,  which,  united  with  the  cents, 
(*048,)  give  4  cents  8  mills  for  the  interest  of  $  1  for  9 
months  and  18  days. 


IT  83.  INTEREST.  ^  161 

^     1.  What  will  be  the  interest  on  $  1  for  5  months  6  days  ? 

—  6  months  12  days  ?  7  months  ?  8  months 

24  days?  9  months   12  days  ?  10  months?  

11   months  6   days?     12  months   18  days?    15 

months  6  days  ?  *  16  months  ?  -«^^ 


Odd  days.'    2.  What  is  the  interest  of  $  1  for  13  months 

16  days  ? 

The  cents  will  be  6,  and  the  mills  5,  for  the  odd  month, 
and  2  for  2  times  6  rz:  12  days,  and  there  is  a  remainder  of 
4  days,  the  interest  for  which  will  be  such  part  of  1  mill  as  4 
days  is  part  of  6  days,  that  is,  |  =  f  of  a  mill.     Ans.  '06 7§. 

Si 3,  What  will  be  the  interest  of  $  1  for  1  month  8  days? 

2  months   7  days?  3  months   15  days?   4 

months  22  days  ?  — —  5  months  11  days  ?  6  months 

17  days?  7  months  3  days  ?  8  months  11  days? 

9  months  2  days  ?  10  months  15   days  ?    

(11  months  4  days  ?  1  12  months  3  days  ? 

Note,  If  there  is  no  odd  months  and  the  number  of  days  he 
less  than  6 y  so  that  there  arc  no  7nills^  it  is  evident,  a  cipher  must 
be  put  in  the  place  of  mills ;  thus,  in  the  last  example, — 12 
months  3  days, — the  cents  will  be  '06,  the  mills  0,  the  3 
days  ^  a  mill.  Ans.  '060^. 

4.  What  will  be  the  interest  of  $  1  for  2  months  1  da^  ? 

4  months  2  days  ?     6  months  3  days  ?     8 

months  4  days  ?     10  months  5  days  ? for  3  days  ? 

for  1   day  ? for  2  days  ?      for  4   days  ? 

for  5  days  ? 

5.  What  is  the  interest  of  $5643  for  S  months  5  days  ? 
The  interest  of  $  1,  for  the  given  time,  is  '040|- ;  therefore, 

^)  and^)  $56'13    principal. 

'040f  interest  of  $  1  for  the  given  time, 

2245*20    interest  for  8  months. 
2806    interest  for  3  days. 
1871    interest  for  2  days. 


2'29197,  Ans.  $2'291. 

'''■^■' 
5  days  =  3  days  -j-  2  days.     As  the  multiplicand  is  talcen 

Otice  for  every  6  days,  for  3  days  take  h  for  2  days  take  |, 


163  INTfiRES'f.  it  83. 

of  the  multiplicand.  ^  -|-  4^  z=  |.  So  also,  if  the  odd  days 
be  4  =:  2  days  -{-  2  days,  take  ^-  of  the  multiplicand  tioice ; 
for  1  day,  take  -J. 

Note,  If  the  sum  on  which  interest  is  to  be  cast  be  less 
than  $  10,  the  interest,  for  any  number  of  days  less  than  6, 
will  be  less  than  1  cent ;  consequently,  in  busmess,  if  the  sum 
be  less  than  $  10,  such  days  need  not  be  regarded. 

From  the  illustrations  now  given,  it  is  evident, — To  find  the 
interest  of  any  sum  in  federal  inonei/j  at  6  per  cent,,  it  is  only 
necessary  to  multiply  the  principal  by  the  interest  of  $  1  for 
the  given  time,  found  as  above  directed,  and  written  as  a 
decimal  fraction,  rememhering  to  point  otF  as  many  pl-aces 
for  decimals  in  the  product  as  there  are  decimal  places  in 
both  the  factors  counted  together. 

EXABIPLES    FOR  PRACTICE. 

6.  What  is  the  interest  of  $  8749  for  1  year  3  months  ? 

Ans,  $6*533. 
»  7.     Interest  of  $116,08  for  11  mo.  19  days?  $6*751, 

8 of  $  200  for  8  mo.  4  days  ?  $  8432. 

^  9 of  $  0*85  for  19  mo.  ?  $  '08. 

10 of  §8'50  for  1  vear  9  mo.  12  days  ?     $'909. 

f  il of  $675  for  1  mo.  21  days  r  $  5'737. 

i2 of  $8673  for  10  days?  $14*455. 

13 of  $0*73  for  10  mo.'?  $  *036. 

14./. of  $  93  for  3  days  ?         )      Note.     The  inte- 

♦  15 of  $  73*50  for  2  days  ?    ^  rest  of  $  1  for  6  days 

16 of  $180*75  for  5  days?  f  being  1  mill, the dol- 

17 of  $15000  for  1  day?  )  lars  themselves  ex- 
press the  interest  in  mills  for  six  days,  of  which  we  may  take 
parts. 

Thus,  6  )  15000  mills, 

2*500,  that  is,  $2^50,  Arts,  to  the  last. 

When  the  interest  is  required  for  a  large  number  of  years, 
it  will  be  more  convenient  to  fmd  the  interest  for  one  year, 
and  multiply  it  by  the  number  of  years ;  after  which  find 
the  interest  for  the  months  and  days,  if  any,  as  usual. 

18.  What  is  the  interest  of  $  1000  for  120  years  ? 

Ans.  $7200 

19.  What  is  the  interest  of  $520*04  for  30  years  and 
6  months?  *  Ans.  $951*673. 


1r  83,  84.  INTEREST.  i63 

20.  What  is  the  interest  on  $400  for  10  years  3  months 
and  6  days  ?  Aiis,  $  246*40. 

21.  What  is  the  interest  of  $220  for  5  years?     for 

12  years  ?     50  years  ?  A^is.  to  last,  $660. 

f  22.  What  is  the  amount  of  $  86,  at  interest  7  years  ? 

Alls.  $12242. 

23.  What  is  the  interest  of  36  i3 .  9  s.  6|-  d.  for  1  year  ? 

Reduce  the  shillings,  pence,  &c.  to  the  decimal  of  a  pound, 
by  inspection,  (*[T76;)  then  proceed  in  all  respects  as  in 
federal  money.  Having  found  the  interest,  reverse  the  ope- 
ration, and  reduce  the  three  first  decimals  to  shillings,  &c., 
by  inspection.    (IF  77. )  Ans.  2  dC .  3  s.  9  d. 

I  24.  Interest  of  36  i2 .  10  s.  for  lS*mo.  20  days  ?   Ans,  3  £ . 
8  s.  U  d.     Interest  of  95  <£ .  for  9  mo.  ?  Aiis,  4  iE .  5  s.  6  d.- 

25.  What  is  the  amount  of  18iS.  12  s.  at  interest  10 
months  3  days  ?  Ans.  Id  £ .  10  s.  9j  d. 

,'  26.  What  is  the  amount  of  100  £ .  for  8  years  ? 

Ans.  148  ^. 

27.  What  is  the  amount  of  400  £.  10  s.  for  18  months? 

A71S.  4^6  £.  10  s.  10  d.  3  q* 

28.  What  is  the  amount  of  640  i3 .  8  s.  at  interest  for  1 
year  ?     for  2  years  6  months  ?     for  10  years  ? 

Ans.  to  last,  1024  i2.  12  s.  9^  d. 

Tf  84.  1.  What  is  the  interest  of  36  dollars  for  8  months, 
at  4^  per  cent.  ? 

Note.  When  the  rate  is  any  other  than  six  per  cent.,  first 
find  the  interest  at  6  per  cent.,  tlten  divide  the  interest  so 
found  by  such  part  as  the  interest,  at  the  rate  required,  ex- 
ceeds or  falls  short  of  the  interest  at  6  per  cent.,  and  the 
quotient  added  to,  or  subtracted  from  the  interest  at  6  per 
cent.,  as  the  case  may  be,  will  give  the  interest  at  the  rate 
required. 


4^  per  cent,  is  |-  of  6  per  cent.  ;  therefore, 
from  the  interest  at  6  per  cent,  subtract  j^ ; 
the  remainder  will  be  the  interest  at  4^  per 
cent. 
VQ8  Ans. 

2.  Interest  of  $64<81  for  18  mo.,  at  5  per  ct.?  A7is.  $441. 

3 of  $500  for  9  mo.  9  days,  at  8 per  ct.?    $31*00 

4 of  ^6242  fori  mo.  20  days,  at  4  per  ct.?  $*345 


164  INTEREST.  IT  84 j  85 

6,  Interest  of  $  85  for  10  mo.  15  days,  at  12 J  percent.  ? 

Ans,  $9'295, 
I  6.  What  is  the  amount  of  $  53  at  10  per  ct.  for  7  mo.  ? 

Ans.  $56^091 

The  timey  rate  per  cent,  and  amount  given^  to  find  the  principaL 

•57fr  85.    1.  What  sum  of  money,  pT\t  at  interest  at  6  per 
cent,  will  amount  to  $61^02,  in  1  yedi  4  months? 

The  amount  of  $  1,  at  the  given  rate  and  time,  is  $  1*08 ; 
hence,  $61^02  --  $  I'OS  =  56'50,  the  principal  required; 
that  is, — Find  the  amount  of  $1  at  the  given  rate  and  tbne^  by 
which  divide  the  given  amount  j  the  quotient  will  he  the  priKci- 
pal  required.  '  Ans.    $  56*50. 

2.  What  principal,  ot  8  per  cent.,  in  1  year  6  months,  will 
amount  to  $  8542  ?  Ans.  $  76. 

3.  What  principal,  at  6  per  cent.,  in  11  months  9  vlays, 
will  amount  to  $  99*311  ? 

Note.  The  interest  of  $  1,  for  the  given  time,  is  *056^  ; 
but,  in  these  cases,  when  there  are  odd  days,  instead  of 
writing  the  parts  of  a  mill  as  a  common  fraction,  it  will  be 
more  convenient  to  write  them  as  a  decimal,  thus,  *0565 ; 
that  is,  extend  the  decimal  to  four  places.  Ans.  $  94. 

4.  A  factor  receives  $  988  to  lay  out  after  deducting  his 
commission  of  4  per  cent. ;  how  much  will  remain  to  be 
laid  out  ? 

It  is  evident,  he  ought  not  to  receive  commission  on  his 
oum  money.  This  question,  therefore,  in  principle,  does  not 
differ  from  the  preceding.  "* 

Note.  In  questions  like  this,  where  no  respect  is  had  to 
time,  (1\  81,  ex.  4,  note,)  add  the  rate  to  $  1.  Ajis.  $  950. 
*  5.  A  factor  receives  $  1008  to  lay  out  after  deducting 
his  commission  of  5  per  cent. ;  w^hat  does  his  commission 
amount  to  ?  A7is.  $  48. 


Discount.  6.  Suppose  I  owe  a  man  j$  397*50,  to  be  paid 
in  1  year,  without  interest,  and  I  wish  to  pay  him  now ;  how 
much  ought  I  to  pay  him  w^hen  the  usual  rate  is  6  per  cent  ? 

I  ought  to  pay  him  such  a  sum  as,  put  at  interest,  would, 
in  1  year,  amount  to  $  397*50.  The  question,  therefore, 
does  not  differ  from  the  preceding.  Ans.  $  376. 

Note.     An  allowance  made  for  the  payment  of  any  sum 


V  85,  86.  INTEREST.  166 

of  money  before  it  becomes  due,  as  in  the  last  example,  i« 
called  Discount. 

The  sum  which,  put  at  interest,  would,  in  the  time  and 
at  the  rate  per  cent,  for  which  discount  is  to  be  made,  amouiit 
to  the  given  sura,  or  debt,  is  called  the  jnesent  worth. 

7.  What  is  the  present  worth  of  $,  S34,  payable  in  1  year 
7  months  and  6  d-ays,  discounting  at  the  rate  of  7  per  cent,  r 

Ans.  §750. 

8.  What  is  the  discount  on  $321^63,  due  4  years  hence, 
discountiog  at  the  rate  of  6  per  cent.  ?  Ans.  $  62^26. 

9.  How  much  ready  money  must  be  paid  for  a  note  of 
$  18,  due  15  months  hence,  discounting  at  tbe  rate  of  6  per 
cent.  ?  Ans.  $  16'744. 

10.  Sold  goods  for  $  650,  payable  one  half  in  4  months, 
and  the  other  half  in  8  months ;  what  must  be  discounted 
for  present  payment  ?  Ans.  $lS'";fX  A 

11.  What  is  the  present  worth  of  §  56^20,  jiay able  in  1 

year  8  months,  discounting  at  6  per  cent.  ?     at  4J  per 

cent.  ?     at  5  per  cent.  ?     at  7  per  cent.  ?     at 

7^  per  cent  ?     at  9  per  cent.  ? 

Am.  to  the  last,  $  48*869. 


The   time^  rate  per  cent.^  and  interest  being  g'wen^  to  find  th^ 
principal. 

•f  86,  1.  What  sum  of  money,  put  at  interest  IG  months, 
will  gain  §  10*50,  at  6  per  cent.  ? 

$1,  at  the  given  rate  and  lime,  will  gain  *08;  hence, 
$10*50-7-  $*08=  $131*25,  the  principal  rec^jired ;  that 
is, — Find  the  interest  of  $  1,  al  the  given  rate  and  twie^  by 
which  divide  the  given  gain,  or  interest ;  the  quotient  will  be  the 
principal  required.  Ans.  $  131*25. 

2.  A  man  paid  $  4*52  interest,  at  the  rate  of  6  per  cent, 
ttt  the  end  of  1  year  4  months ;  what  was  the  principal  ? 

Ans.  $56*50. 

3.  A  man  received,  for  interest  on  a  certain  note,  at  the 
end  of  1  year,  $  20  ;  what  was  the  principal,  allowing  the 
rate  to  hiave  been  6  per  cent,  ?  -4??^.  $  333*333|. 


166  ,  INTEREST.  IT  87,  88* 

The  principal^  interest^  and  time  being  given^  to  find  the  rate 
per  cent. 

ir  87.  1.  If  I  pay  $3'78  interest,  for  the  use  of  $36 
for  1  year  and  6  months,  ^vhat  is  that  per  cent.  ? 

The  interest  on  $  36,  dXoneper  cent,  the  given  time,  is  $  *o4 ; 
hence,  $  3'78  -f-  $  '54  =  '07,  the  rate  required  ;  that  is, — 
Find  the  interest  on  the  given  sum,  at  1  per  cent.  Jar  the  given 
time,  by  which  divide  the  given  interest ;  the  quotient  mill  be 
the  rate  at  which  interest  was  paid,  Ans,  7  per  cent. 

'  f  2.  If  I  pay  $  2'34  for  the  use  of  $  468,  1  month,  what  is 
4he  rate  per  cent.  ?  Ans,  6  per  cent 

3.  At  $46'80  for  the  use  of  $520,  2  years,  what  is  that 
per  cent.  ?  Ans,  4^  per  cent 


The  prices  at  which  goods  are  bought  and  sold  being  given^  to 
find  the  rate  per  cent,  of  gain  or  loss. 

^  88.  i.  If  I  purchase  wheat  at  $  I'lO  per  bushel,  and 
sell  it  at  $  1'37^  per  bushel,  w]^at  do  I  gain  per  cent.  ? 

This  question  does  not  differ  essentially  from  those  in  the 
foregoing  paragraph.  Subtracting  the  cost  from  the  price 
at  sale,  it  is  evident  I  gain  27^  cents  on  a  bushel,  that  is, 
^  of  the  first  cost  '^  =  '25  per  cent.,  the  Answer,  That  is, 
— Make  a  common  fraction^  writing  the  gain  or  loss  for  the  nnmcra- 
tor  J  and  the  price  at  which  the  article  was  bought  for  the  de- 
nominator;   then  reduce  it  to  a  decimal, 

2.  A  merchant  purchases  goods  to  the  amount  of  $  550  ; 
what  per  cent,  profit  must  he  make  to  gain  $  66  ? 

Ans,  12  per  cent 

3.  What  per  cent,  profit  must  he  make  on  the  same 

purchase  to  gain  $  38*50  ?     to  gain  $  24'75  ?     to 

gain  $2'75? 

Note,  The  last  gain  gives  for  a  quotient  '005,  which  is  ^ 
per  cent.  The  rate  per  cent,  it  must  be  recollected,  (If  81, 
note,)  is  a  decimal  carried  to  two  places,  or  hundredths ;  all 
decimal  expressions  lower  than  hundredths  are  parts  of  1 
per  cent 

"  \4.  Bought  a  hogshead  of  rum,  containing  114  gallons,  at 
96  cents  per  gallon,  and  sold  it  again  at  $  1'0032  per  gal- 
lon ;  what  was  the  whole  gain,  and  v/hat  was  the  gain  per 
cent?  4       J  $4'924,  whole  gain. 

)     44-  gain  per  cent. 


IT  88,  89.  INTEREST.  167 

5.  A  merchant  bought  a  quantity  of  tea  for  $  365,  which, 
proving  to  have  been  damaged,  he  sold  for  $33245;  what 
did  he  lose  per  cent.  ?  Ans.  9  per  cent. 

6.  If  I  buy  cloth  at  $2  per  yard,  and  sell  it  for  $2'50 
per  yard,  what  should  I  gain  in  laying  out  $  100  ? 

Am,  $25. 

7.  Bought  indigo  at  $  1^20  per  lb.,  and  sold  the  same  at 
90  cents  per  lb. ;  what  was  lost  per  cent.  ?     Ans,  25  per  cent. 

.  8.  Bought  30  hogsheads  of  molasses,  at  $  600  ;  paid  in 
duties  $  20-66  ;  for  freight,  $  40'78 ;  for  porterage,  $  6^05, 
and  for  insurance,  $  30*84  :  if  I  sell  them  at  $  26  per  hogs- 
head, how  much  shall  1  gain  per  cent.  ?  ^725.11^695per  cent. 


The  principal,  rate  per  cent.^  and  interest  being  given,  to  find 
the  time, 

^  89.  1.  The  interest  on  a  note  of  $  36,  at  7  per  cent., 
v/as  $  3'78 ;  what  was  the  time  ? 

The  interest  on  $  36,  1  year,  at  7  per  cent,  is  $  2'52 ; 
hence,  $  3'78  -7-  $  2'52  =  1^5  years,  the  time  required ;  that 
is, — Find  the  interest  for  1  year  on  the  principal  given,  at  the 
given  rate,  by  which  divide  the  given  interest ;  the  quotient  will 
be  the  time  required,  in  years  and  decimal  parts  of  a  year  ;  the 
latter  may  then  be  reduced  to  months  and  days. 

Ans,  1  year  6  months, 
(2,  If  $31^71  interest  be  paid  on  a  note  of  $226'50, 
what  was  the  time,  the  rate  being  6  per  cent.  ? 

Ans,  2 '33-^  :=  2  years  4  months, 

3.  On  a  note  of  $  600,  paid  interest  $  20,  at  8  per  cent. ; 
what  was  the  time  ? 

Ans.  *416  -]-  z=.  5  months  so  nearly  as  to  be  called  5,  and 
would  be  exactly  5  but  for  the  fraction  lost. 
'  4.-  The  interest  on  a  note  of  $  217^25,  at  4  per  cent,  was 
$  28'242 ;  what  was  the  time  ?  Ans,  3  years  3  months. 

Note,  When  the  rate  is  6  per  cent.,  we  may  divide  the 
interest  by  ^  the  principal,  removing  the  separatrix  two 
places  to  the  left,  and  tlie  quotient  will  be  tlie  answer  in 
months* 


168  INTEREST.  IT  90. 


To  find  the  interest  due  an  notes^  S^c.  when  partial  payments 
have  been  iuade. 

^  90-  In  Massachusetts  the  law  provides,  that  payments 
shall  he  applied  to  keep  down  the  interest,  and  that  neither 
interest  nor  payment  shall  ever  draw  interest.  Hence,  if  the 
payment  at  any  time  exceed  the  interest  computed  to  the 
same  time,  that  excess  is  taken  from  the  principal ;  but  if 
the  payment  be  less  than  the  interest,  llie  principal  remains 
unaltered.  Wherefore,  we  have  this  Rule  : — Compute  the 
interest  to  the  first  time  when  a  payment  w^as  made,  which, 
either  alone,  or  together  with  the  preceding  payments,  if 
any,  exceeds  the  interest  t/ien  due;  add  that  interest  to  the 
principal,  and  from  the  sum  subtract  the  payment,  or  the 
$um  of  the  payments,  made  within  the  time  for  which  the 
interest  was  computed,  and  the  remainder  will  be  a  new 
principal,  with  which  proceed  as  with  the  first. 

1.  For  value  receioedj  I  promise  to  pay  James  Conant,  or 
order^  one  hundred  sixteen  dollars  sixty-six  cents  and  six  mills^ 
with  interest.     May  1,  1822. 

g  116,686.  Samuel  Rood. 

On  this  note  were  the  folio  win  cr  endorsements  : 


Dec.  25,  1822,  received  $  l&^QQQ  ^ 

July  10,   1823,  $    1-666 

Sept.   1,   1824,  $    5'000 

June  14,  1825,  ^33*333 

April  15,  1826,  $62^000 

What  was  due  August  3,  1827  r  Ans.  $  23^775, 


Note,  In  finding  the 
times  for  computing  the 
interest,  consult  IT  40. 


The  first  principal  on  interest  from  May  1,  1822,    $  116*666 
Interest  to  Dec.  25,  1822,  time  of  the  first  pay- 
ment,  (7  m.onths  .24  days,)  -         -         -  4*549 

Amount,  $  121*215 
Payment,  Dec.  25,  exceeding  interest  then  due,  16*666 

Remainder  for  a  new  principal,        -         -         -         104*549 
Interest  from  Dec.  25,  1S22,  to  June   14,  1825, 

(29  months  19  days,)  -         .         -         .  15*490 

Amount  carried^ forward,    $  120*039 


IT  95,  91.  COMPOUND   INTEREST*  169 

Amount  broujrht  fonrard,    $120'039 
Payment,  July  19,  1823,  Icj-s  than  interest 

thendwe,  -        -         -        -         $    1*666 

Payment,  Sept.  1,  1S24,  less  than  interest 

then  due,  ...         -  5*000 

Payment,  June  14,  1825,  exceeding  in- 
terest then  due,  -        .        -  33*333 

$39^09 

Remainder  for  a  new  principal,  (June  14,  1825,)        80^040 
Interest  from  June   14,   1825,  to  April  15,  1823, 

(10  months  1  day,)  .         *        .        .  4^015 

Amount,  $  fa4'055 
Pa}Tnent,  April  15,  1825,  exceeding  intereht  then 

due, 62*000 

Ren^ainder  for  a  new  principal,  (April  15, 1826,)     $  22'055 
Interest  due  Aug.  3,  1827,  from  April  15,  1826, 

(15  months  18  days,)  .        «        -        .  1<720 

Balance  due  Aug  3,  1827,  -         -         $  23^775 

2.  For  value  received^  T promise  to  pay  Jameq  Lowell,  qt 
ctdeTy  eight  hundred  sixty-set eii  dollars  aiid  thirty-three  cents, 
with  interest.     Jan,  6.  1820, 

$867*33.  Hiram  Simson;* 

On  this  note  were  the  follmving  endorsements,  vU» 
April  16,  1823,  received  $136*44. 
April  16,  1825,  received  $319. 
Jan.     1,  1826,  received  $518*68. 

What  remained  due  July  11,  1827  ?         Am*    $215*li),i 


COMPOUND  INTEREST. 

IT  n»  A  promises  to  pay  B  $256  in  3  yean,  ^^th  in- 
terest annually;  but  at  the  end  of  1  y^<jar,>not  fiacing  it  con- 
veaient  to  pay  the  interest,  he  consents  to  pny  interest  on 
the  interest  from  that  time,  the  same  as  on  ths  prin  :ipAl. 

Note,  Simple  interest  is  that  which  is  allowed  for  the 
principal  only ;  compoimd  interest  is  that  wlvich  is  allowed 


170  COMPOUND    INTEREST.  IT  91, 

for  both  principal  and  interest^  when  the  latter  is  not  paid  at 
the  time  it  becomes  due. 

Compound  interest  is  calculated  by  adding  the  interest  to 
the  principal  at  the  end  of  each  year,  and  making  the  cunoimt 
the  principal  for  the  next  succeeding  year. 

1.  What  is  the  compound  interest  of  $256  for  3  years, 
at  6  per  cent.  ? 

$  256  given  sum,  or  first  principal. 
'06 


15 
256 


•'36  interest,    >  .    i        u  j  ^      ^^v 
I'OO  principal,  5  *<'''^^'''^*^"^*^g^*« 


271'36  amount,  or  principal  for  2d  year. 
'06 


10'2816  compound  interest,  2d  year,  >  addCvl  to- 
271 '36      principal,  do.       )  g<-'ther. 

287'6416  amount,  or  principal  for  3d  year. 
'06 


17'25846    compound  interest,  3d  year,  )  added  to- 
28T'641        principal,  do.       ]  gether. 

304'899        amount. 

256  first  principal  subtracted. 

Ans,  $4S'899        compound  interest  for  3  years. 

/  2.  At  6  per  cent.,  what  will  be  the  compound  interest,  and 

wiiat  the  amount,  of  $  1  for  2  year?  ?     what  the  amount 

for  3  years  ?     for  4  years  ? for  5  years  ?     for 

6  years  ?     for  7  years  ? for  S  years  ? 

Ans.  to  the  last,  $  1'593-f-. 

It  is  plain  that  the  amount  of  $  2,  for  any  given  time,  will 
be  2  times  as  much  as  the  amount  of  $  1 ;  the  amount  of 
$3  will  be  3  times  as  much,  &c. 

Hence,  we  may  form  the  amounts  of  $  1,  for  several  years, 
into  a  table  of  multipliers  for  finding  the  amount  of  any  sum 
tor  the  same  time. 


fr  91. 


COMPOUND    INTEREST. 


171 


TABLE, 

Showing  the  amount  of  $  1,  or  l£.,  &c.  for  any 
years,  not  exceeding  24j  at  the  rates  of  5  and 
compound  interest. 


Tears. 

1 

2 

3 

4 

6 

6 

7 

8 

9 
10 
11 
12 


5  per  cent. 

1'05 
14025 
1457G2  + 

1'21550  4- 
1*27628  + 
1*34009  + 
1*40710  + 
1*47745  + 
1*55132  + 
1*62889  + 
1*71033  + 
1*79585  + 


6  per  cent. 

1*06 
1*1236 
1*19101  + 
1*26247  + 
1*33822  + 
1*41851  + 
1*50363  + 
1*59384  + 
1*68947  + 
1^79084  + 
1*89829  + 
2*01219  + 


Years. 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


5  per  cent. 
1*88564  + 
1*97993  + 
2*07892  + 
2*18287  + 
2*29201  + 
2*40661  + 
2*52695 
2*65329  + 
2*78596  + 
2*92526  + 
3*07152  + 
3*22509  + 


number  ot 
6  per  cent 

6  per  cent. 

2*13292  + 
2*26090  + 
2*39655  + 
2*54035  + 
2*69277  + 
2*85433  + 
3*02559  + 
3*20713  + 
3*39956  + 
3*60353  + 
3^81974  + 
4*04893  + 


Note  1.  Four  decimals  in  the  above  numbers  will  be  suf- 
ficiently accurate  for  most  operations. 

Note  2.  When  there  are  months  and  days,  you  may  first 
find  the  amount  for  the  years^  and  on  that  amount  cast  the 
interest  for  the  months  and  days ;  this,  added  to  the  amount^ 
will  give  the  answer. 

3.  What  is  the  amount  of  $600*50  for  20  years,  at  5  per 
cent,  compound  interest .''  at  6  per  cent.  ? 

$  1  at  5  per  cent.,  by  the  table,  is  $  2*65329  ;  therefore, 
2*65329  X  600*50  =z  $  1593*30  +  Arts,  at  5  per  cent. ;  and 
3*20713  X  600*50  z=  $  1925*881  +  Aiis.  at  6  per  cent. 

|4.  What  is  the  amount  of   $  40*20  at  6  per  cent,  com- 
pound interest,  for  4  years  .^  for  10  years  ?  for  18 

years  ?  for  12  years  ?    for  3  years  and  4  months  ^ 

for  24  years,  6  months,  and  18  days  ? 

Am.  tolasi^  $  168*137. 

Note.  Any  sum  at  compound  interest  will  double  itself 
in  11  years,  10  months,  and  22  days. 

From  what  has  now  been  advanced  we  deduce  the  fol- 
lowing general 

RULE. 

I.  To  find  the  interest  when  the  time  is  1  year,  or,  to  find  /A# 
rate  per  cent,  on  any  sum  of  mwiey^  without  respect  to  timCy  a» 


\U2,  COMPOUND    INTEREST.  ^91. 

the  premium  for  insurance^  commission^  &c., — Multiply  tLe 
principal,  or  given  sum,  by  the  rate  per  ceut.,  written  as  a 
decinial  fraction ;  the  product,  remembering  to  point  otf  a?- 
many  places  for  decimals  as  there  are  decimals  in  both  the 
factors,  will  be  the  interest,  &c.  required. 

II.  IVlten  there  are  months  and  (lays  in  the  given  iime^  to  find 
th<i  interest  on  any  sum  of  money  at  6  per  cent,^ — Multiply  the 
principal  by  the  interest  on  $  1  for  the  given  time,  found  by 
inspection,  and  the  product,  as  before,  \vill  be  the  interest 
required. 

III.  To  find  the  interest  on  $1  at  6  per  cent,,  for  any  given 
time^  by  inspection^ — It  is  only  to  consider,  that  the  cents  ^viij 
be  equal  to  half  uie  greatest  even  number  of  the  months ; 
and  the  mills  will  be  5  for  the  odd  month,  (if  there  be  one,) 
and  1  for  every  6  days. 

ly.  If  the  sum  given  be  in  pounds,  shiUings.  pence  andfar^ 
things, — Reduce  the  shillings,  &:c.  to  the  decimal  of  a  pound, 
by  inspection,  (tl  76  ;)  then  proceed  in  all  respects  as  in 
federal  money.  Having  fouud  the  interest,  the  decimal  part, 
by  reversing  tlie  operation,  may  be  reduced  back  to  shillings, 
pence  and  farthings. 

y.  If  the  interest  reqmred  he  at  any  other  rate  than  6  pet 
cent,,  {if  there  be  months,  or  months  and  days,  in  the  given  time,) 
— First  find  the  interest  at  6  per  cent. ;  then  divide  the  in- 
terest so  found  by  such  part  or  pirts,  as  the  interest,  at  the 
rate  requirF'l,  exceeds  or  falls  short  of  the  interest  at  6  per 
cent.,  and  the  qr.otient,  or  quotients,  added  to  cr  subtracted 
from  the  interest  at  6  per  cent.,  as  the  case  may  require,  will 
give  the  interest  at  the  rate  required. 

Note,  The  interest  on  any  number  of  dollars,  for  6  days^ 
at  6  percent.,  is  readily  found  by  cutting  oif  the  unit  or  right 
hand  P.gure  ;  those  at  the  left  hand  will  show  the  iuterest  in 
cent^  for  6  days. 

EXA3IPI.es    FOn    PRACTICE- 

•  1.  What  is  the  interest  of  $  1600  for  1  vear  and  3  months  ? 

Ans,    $120. 
2.  What  Is  tlic  interest  of  $  £*S11,  for  1  year  1 1  months  ? 

A?is.  $'668. 
f  3.  What  is  the  interest  of  $2'29,  for  1  month  19  days, 
at  3  per  cent.?  Ans,    $'009. 

/  4.  What  is  .Uiciiiterest  of  $  18,  for  2  yeais  14  days  at  7 
pec  Q*^iXi,  I  ^is.   §  2'56a,, 


IT  91.  SUPPLEMENT    TO    INTEREST.  173 

5.  What  is  the  interest  of  $17^68,  for  11  months  28 
days?  Ans,    $1*054. 

6.  What  is  the  interest  of  $  200  for  1  day  ?  2  days  ? 

3  days  ?  4  days?  5  days  ? 

Ans,  for  5  days,  $  0466. 
/7.  What  is  the  interest  of  half  a  mill  for  567  years? 

Arts.    $0^017. 
%.  What  is  the  interest  of  $  81,  for  2  years  14  days,  at  ^ 

per  cent.  ?  f  per  cent.  ?  |  per  cent.  ?  2  per 

cent.  ?  3  per  cent.  ?  4-J-  per  cent.  ?  5  per 

cent.  ?  6  per  cent.  ?  7  per  cent.  ?  7^  per 

cent.  ?  8  per  cent.  ?    *  9  per  cent.  ?  10  per 

cent.  ?  12  per  cent  ?  i  12J-  per  cent.  ? 

Ans,  to  last,  $20^643. 

9.  What  is  the  interest  of  9  cents  for  45  years,  7  months, 
11  days?  Ans.    $0'245. 

10.  A's  note  of  $  175  was  given  Dec.  6,  1798,  on  which 
was  endorsed  one  year's  interest ;  what  was  there  due  Jan. 
1,  1803? 

Note.     Consult  ex.  16,  Supplement  to  Subtraction  of  Com- 

found  Numbers.  Aiis.    $207'22. 

11.  B's  note  of  $  56^75  was  given  June  6,  1801,  on  inter- 
eist  after  90  days;  what  was  there  due  Feb.  9,  1802  ? 

Ans.    $5849. 
1 12.  C'snote  of  $365^37  was  given  Dec.  3,  1797;  June 
7,  1800,  he  paid    $9746;  what  was  there  due   Sept.  11, 
1800?  Ans.    $328*32. 

13.  Supposing  a  note  of  $317*92,  dated  July  5,  1797,  on 
which  were  endorsed  the  follov/ing  payments,  viz.  Sept.  13, 
1799,  $208*04;  March  10,  1800,  $76;  what  was  there 
due  Jan.  1,  1801  ?  Ans.   $83*991. 


SUPPLEMENT   TO  INTEREST. 

QUESTIONS. 

1.  What  is  interest  ?     2.  How  is  it  computed  ?     3.  What 

is   understood   by  rate  per  cent.  ?     4.  by  principal  ? 

5.  by  amount  ?     6.  by  legal  interest  ?     7.  

by  commission?     8.  insurance?     9.  premium? 

10.  policy?      11.  stock?      12.  What  is  undep- 

•tood  by  stock  being  at  pari     13.  abore  par?     14 


174  6urrt.EiviErtJT  to  interest.  it  ti^ 

below  pur  ?  15.  The  rale  per  cent,  is  a  decimal  car- 
ried to  how  many  places  ?  IG.  What  are  decimal  expres- 
sions/omcr  than  hundredths  r  17.  How  is  interest,  (when 
the  time  is  1  year,)  commission,  insurance,  or  any  thing  else 
rated  at  so  much  per  cent,  without  respect  to  time,  found  ? 
18.  When  the  rate  is  1  per  cent.,  or  less,  how  may  the  ope- 
ration be  contracted.^     19.  How  is  the  interest  on  $  1,  at 

6  per  cent,  for  any  given  time,  found  by  inspection  ?  20. 
How  is  interest  cast,  at  6  per  cent.,  when  there  are  months 
and  days  in  the  given  time  ?  21.  When  the  given  time  is 
less  tlian  6  days,  how  is  the  interest  most  readily  found  ? 
22.  If  the  sum  given  be  in  pounds,  shillings,  &c.,  how  is  in- 
terest cast?  23.  When  the  rate  is  any  other  than  6  per 
cent,  if  there  be  months  and  days  in  the  given  time,  how  is 
the  interest  found  ?  24.  What  is  the  rule  for  casting  interest 
on  notes,  &c.  when  partial  payments  have  been  made,  and 
what  is  the  principle  on  which  the  rule  is  founded  ?  25. 
How  may  the  principal  be  found,  the  time,  rate  per  cent, 
and  amount  being  given  ?  26.  What  is  understood  by  dis- 
count 1  27.  by  present  worth  ?  28.  How  is  the  prin- 
cipal found,  the  time,  rate  per  cent.,  and  interest  being  given  ? 
29.  How  is  the  rate  per  cent,  of  gain  or  loss  found,  the 
prices  at  which  goods  are  bought  and  sold  being  given  ?  30- 
How  is  the  rate  per  cent,  found,  the  principal,  interest,  and 
time  being  given  ?  31.  How  is  the  time  found,  the  princi- 
pal, rate  per  cent,  and  interest  being  given  ?     32.  What  is 

simple  interest.^     33.  compound  interest?     34.  How 

is  compound  interest  computed  ? 

EXERCISES, 

1.  What  is  the  interest  of  $273^51  for  1  year  10  days,  at 

7  percent.?  Ans.    $19*677. 

2.  What  is  the  interest  of  $486  for  1  year,  3  months,  19 
days,  at  8  per  cent.  ?  Am,    $  50*652. 

3.  D's  note  of  $203*17  was  given  Oct.  5,  1808,  on  inter- 
est after  three  months;  Jan.  5,  1809,  he  paid  $50;  what 
was  there  due  May  2,  1811  ?  Am.    $  174^53. 

4.  E's  note  of  $870*05  was  given  Nov.  17,  1800,  on  in- 
terest after  90  days  ;  Feb.  11,  1805,  he  paid  $186*06;  what 
was  there  due  Dec.  23, 1807  ?  Ans.  $  1041<5a 
^  6.  What  will  be  the  annual  insurance,  at  f  per  cent.,  on 
a  house  valued  at  $  1600  ?  Ans,   %  10. 


IT  91.  SUPPLEMENT    TO    INTEREST.  175 

G.  What  will  be  the  insurance  of  a  ship  and  carp^o,  valued 

at  $  5643,  at  1^  per  cent.  ?  at  f  per  cent.  ? at  -f^ 

per  cent.  ?  at  {^  per  cent.  ?  at  f  per  cent.  ? 

Note,     Consult  fi  82,  ex.  11. 

.4715.  at  f  per  cent.  $  42^322. 

^7.  A  man  having  compromised  with  his  creditors  at  62^ 

cents  on  a  dollar,  what  must  he  pay  on  a  debt  of  $  137'46  ? 

xins.    $85'9>2. 

I  8.  What  is  the  value  of  $  800  United  States  Bank  stock, 

at  112^  per  cent.  ?  Am,    $  900. 

9.  What  is  the  value  of  $  560^75  of  stock,  at  93  per  cent.  ? 

Ans,    $521 '497 

10.  What  principal  at  7  per  cent,  will,  in  9  months  18  days, 
amount  to  $  422'40  ?  Ans,    $  400^ 

11.  What  is  the  present  worth  of  $  426,  payable  in  4 
years  and  12  days,  discounting  ai  the  rate  of  5  per  cent.  ? 

In  large  sums,  to  bring  out  the  cents  correctly,  it  w*ill 
sometimes  be  necessary  to  extend  the  decimal  in  the  divisof 
to  ^ve  places.  Aiis,   $  354'o06. 

12.  A  merchant  purchased  goods  for  $250  ready  money, 
and  sold  them  again  ior  $300,  payable  in  9  months;  what 
did  he  gain,  discounting  at  6  per  cent.  ?  A?is.    $37^081. 

f  13.  Sold  goods  for  $3120,  to  be  paid,  one  half  in  3 
months,  and  the  other  half  in  6  months ;  what  must  be  dis- 
counted for  present  payment  ?  Ans,  68'492. 

14.  The  interest  on  a  certain  note,  for  1  year  9  months, 
was  $  49^875  ;  what  was  the  principal  ?  Ans.    $  475. 

15.  What  principal,  at  5  per  cent.,  in  16  months  24  days, 
will  gain  $  35  ?  Ans.    $  500. 

16.  If  I  pay  $15*^50  interest  for  the  use  of  $500,9 
months  and  9  days,  what  is  the  rate  p'3r  cent.  ? 

f  17.  If  I  buy  candles  at  $467  per  lb.,  and  sell  tliem  at 
20  cents,  what  shall  I  gain  in  laying  out  $  100  ? 

^71*.    $19<76. 

18.  Bought  hats  at  4  s.  apiece,  and  sold  them  again  at  4  8i 
9  d. ;  what  is  the  profit  in  laying  out  100  ^ .  ? 

Ans,  18  ig.  15  s* 

19.  Bought  .37  gallons  of  brandy,  at  $140  per  gallon, 
and  sold  it  for  $  40 ;  what  was  gained  or  lost  per  cent.  ? 

20.  At  4  s.  6  d.  profit  on  1  £,,  how  much  is  gained  in  laying 
out  100  £ .,  that  is,  bow  much  per  cent.  ?     Ans,  22  iS .  10  s* 

•  21.  Bought  cloth  at  $4'48  per  yard ;  how  must  I  sell  il 
to  gain  12;J  per  cent.  ?        ^  Ans.   $  5*04 


176  EQUATION    OF    PAYMENTS.  IT  91,  92 

22.  Bought  a  barrel  of  powder  for  4  iC . ;  for  how  much 
must  it  be  sold  to  lose  10  per  cent.  ?  Ans,  3  iS .  12  9. 

23.  Bought  cloth  at  15  s.  per  yard,  which  not  proving  so 
good  as  I  expected,  I  am  content  to  lose  47^  per  cent. ;  hoMr 
must  I  sell  it  per  yard  ?  Ans,  12  s.  4Jd 
{  24.  Bought  50  gallons  of  brandy,  at  92  cents  per  gallon, 
fcut  by  accident  10  gallons  leaked  out ;  at  what  rate  must  I 
sell  the  remainder  per  gallon  to  gain  upon  the  whole  cost  al 
the  rate  of  10  per  cent.  ?                   Am,    $  1*265  per  gallon. 

25.  A  merchant  bought  10  tons  of  iron  for  $  950 ;  the 
freight  and  duties  came  to  $  145,  and  his  own  charges  to 
$  25  ;  how  must  he  sell  it  per  lb.  to  gain  20  per  cent,  by  it  f 

Ans.  6  cents  per  lb. 


EQUATXOZf  OF  PAimXENTS. 

IT  92.  Equation  of  payments  is  the  method  of  finding  the 
mean  time  for  the  payment  of  several  debts,  due  at  different 
times. 

1.  In  how  many  months  will  $  1  gain  as  much  as  5  dol- 
lars will  gain  in  6  months  ? 

2.  In  how  many  months  will  $  1  gain  as  much  as  $  40 
will  gain  in  15  months  ?  Ans.  600. 

3.  In  how  many  months  will  the  use  of  $  5  be  worth  as 
much  as  the  use  of  $  1  for  40  months  ? 

4.  Bc-rrowed  of  a  friend  $  1  for  20  months ;  afterwards 
lent  my  friend  $  4  ;  how  long  ought  he  to  keep  it  to  becomij 
indemnified  for  the  use  of  the  $1? 

5.  I  have  three  notes  against  a  man ;  one  of  $  12,  due  in 
3  months ;  one  of  $  9,  due  in  5  months  ,•  and  the  other  of 
$  6,  due  in  10  months ;  the  man  wishes  to  pay  the  whole  at 
once;  in  what  time  ought  he  to  pay  it  ? 

$  12  for  3  months  is  the  same  as  $  1  for  36  months,  and 
$  9  for  5  months  is  the  same  as  $  1  for  45  months,  and 
$   6  for  10  months  is  the  same  as  $  1  for  60   months. 

27  141 

He  might,  therefore,  have  $1  141  months,  and  he  may 
keep  27  dollars  ^y  part  as  long ;  that  is,  ^^  =  5  monllii 
6  -}-  days,  Answer, 


IT  93.     ratio;  or  the  relation  of  numbers.       177 

Hence,  To  find  the  mean  time  for  several  payments^ — Hule; 
— Multiply  each  sum  by  its  time  of  payment,  and  divide  the 
sum  of  the  products  by  the  sum  of  the  paijinents^  and  tlie 
quotient  will  be  the  answer. 

Note,  This  rule  is  founded  on  the  supposition,  that  what 
13  gained  by  keeping  a  debt  a  certain  time  after  it  is  due,  is 
the  same  as  what  is  lost  by  paying  it  an  equal  time  before  it 
is  due ;  but,  in  the  first  case,  the  gain  is  evidently  equal  to  the 
interest  on  the  debt  for  the  given  time,  while,  in  the  second 
case,  the  loss  is  only  equal  to  the  discount  of  the  debt  for  that 
time,  which  is  always  less  than  the  interest;  therefore,  the 
rule  is  not  exactly  true.  The  error,  however,  is  so  trifling, 
in  most  questions  that  occur  in  business,  as  scarce  to  merit 
notice. 

6.  A  merchant  has  owing  him  $300,  to  be  paid  as  fol- 
lows :  $50  in  2  months,  $  100  in  5  months,  and  the  rest  in 
8  months ;  and  it  is  agreed  to  make  one  paymeiit  of  the 
whole  :  in  what  time  ought  that  payment  to  be  ? 

Ans,  6  months. 

7.  A  owes  B  $  136,  to  be  paid  in  10  months ;  $  96,  to  be 
paid  in  7  months;  and  $260, to  be  paid  in  4  months:  what 
IS  the  equated  time  for  the  payment  of  the  whole  ? 

Ans.  6  months,  7  days  -{-. 

8.  A  owes  C  $600,  of  whicli  $  200  is  to  be  paid  at  the 
present  time,  200  in  4  months,  and  200  in  8  months  ;  what 
IS  the  equated  time  for  the  payment  of  the  v/hole  ? 

Ans.  4  months- 

9.  A.  owes  B  $  300,  to  be  paid  as  follows  :  ^  in  3  months, 
^  in  4  months,  and  the  rest  in  6  months  :  what  is  tlie  equated 
time  ?  Ans,  4^  months. 

X'v  ^^'^^^'^^ 

/^  OF  THK  \\ 

RATIO  J i^^^^^^^^ 

OR  ^X^r/.,.. 

TT  93.  1.  What  part  of  1  gallon  is  3  quarts?  1  gallon  is 
4  quart«,  and  3  quarts  is  J  of  4  quarts.       An^,  |  of  a  gallon- 

2.  What  part  of  3  quarts  is  1  gallon  ?  1  gallon,  being  4 
quarts,  is  ^  of  3  quarts ;  that  is,  4  quarts  is  1  time  3  (piarta 
and  ^  of  another  time.  Ans.  J  zz=  1  ^ 


nS        RATIO  ;    OR    THE    RELATION    OF    NUMBERS.       M  9^^ 

3.  What  part  of  5  bushels  is  12  bushels  ? 

Fii.ding  what  part  one  number  is  of  another  is  the  same 
as  finding  what  is  called  the  ratio^  or  relation  of  one  number 
to  another;  thus,  the  question,  What  part  of  5  bushels  is  12 
bushels  ?  is  the  same  as  What  is  the  ratio  of  5  bushels  to  12 
bushels  ?     The  Ansxoer  is  J^  zz=  2f . 

Ratioj  therefore,  may  be  defined^the  number  of  times  one 
number  is  contained  in  another;  or,  the  number  of  times  one 
quantity  is  contained  in  another  quantity  of  the  same  kind. 

4.  What  part  of  8  yards  is  13  yards  ?  or,  What  is  the  ratio 
of  8  yards  to  13  yards? 

13  yards  is  ^^-  of  8  yards,  expressing  the  ^\\\doi\  fractimall^ 
If  now  we  perf:)rm  the  division,  w^e  have  for  the  ratio  \^ ; 
that  is,  13  yards  is  1  time  8  yards,  and  |  of  another  time. 

We  have  seen,  (IT  Ib^sicjn^)  that  division  maybe  expressed 
fractionally.  So  also  the  ratio  of  one  number  to  another,  or 
the  part  one  number  is  of  another,  may  be  expressed  frao- 
tionfll},  to  do  which,  make  the  number  which  is  called  the 
partj  w^hether  it  be  the  larger  or  the  smaller  number,  the  nu- 
merator of  a  fraction,  under  which  write  the  other  number  for 
a  denominator.  When  the  question  is.  What  is  the  ratio,  &c. } 
the  number  last  named  is  the  part ;  consequently  it  must  be 
made  the  numerator  of  the  fraction,  and  the  number  first 
named  the  denominator. 

5.  What  part  of  12  dollars  is  11  dollars  ?  or,  11  dollars  is 
what  part  of  12  dollars  ?  1 1  is  the  number  which  expresses 
i\iQ  part.  To  put  this  question  in  the  other  form,  viz.  AVhat 
is  the  ratio^  &.c.  r  let  that  number,  which  expresses  the  part^ 
be  the  number  last  named ;  thus,  What  is  the  ratio  of  12  dol- 
lars to  1 1  dollars  ?  Ans.  \^, 

6.  What  part  of  1  jg .  is  2  s.  G  d.  ?  or,^Tiat  is  the  ratio  of 
1  ie .  to  2  s.  6  d.  ? 

\  £,  z=z  240  pence,  and  2  s.  6  d.  z=.  30  pence ;  hence, 
^^  =z  ^,  is  the  Answer, 

7.  What  part  of  13  s.  6  d.  is  1  iB .  10  s.  ?  or,  What  is  the  ra- 
tio of  13s.  6d.  to  1  £,  10s.?  Ans.  ^<^ 

8.  What  is  the  ratio  of  3  to  5  ?  of  5  to  3  ?  of 

7  to  19  ?  of  19  to  7?  of  15  to  90  ?  of  90  to 

15  ?  of  84  to  160  ?  of  160  to  84  ?  of  615  to 

1107  ?  of  1107  to  615  ?  Ans,  to  the  last,  f 


\ 


V  94.  KULE   OF  THREE.  179 

PROPORTZON; 

OR  * 

THE   RUZS   OF  THKSE. 

IT  94.  1.  If  a  piece  of  cloth,  4  yards  long,  cost  12  dollars, 
what  will  be  the  cost  of  a  piece  of  the  same  cloth  7  yards 
long  ? 

Had  this  piece  contained  twice  the  number  of  yards  of  the 
first  piece,  it  is  evident  the  price  would  have  been  twice  as 
much ;  had  it  contained  3  times  the  number  of  yards,  the 
price  would  have  been  3  times  as  much ;  or  had  it  contained 
only  half  the  number  of  yards,  the  price  would  have  been 
only  half  as  much ;  that  is,  the  cost  of  7  yards  wnl  be  such 
part  of  12  dollars  as  7  yards  is  part  of  4  yards.  7  yards  is 
J  of  4  yards  ;  consequently,  the  price  of  7  yards  must  be  J  of 
Qie  price  of  4  yards,  or  J  of  12  dollars.  {-  of  12  dollars,  that 
is,  12  X  J  =  -^j^-  =r  21  dollars.  Answer. 

2.  If  a  horse  travel  30  miles  in  6  hours,  how  many  miles 
will  he  travel  in  11  hours,  at  that  rate  ? 

11  hours  is  -y-  of  6  hours,  that  is,  11  hours  is  1  time  6 
hours,  and  f  of  another  time  ;  consequently,  he  will  travel,  in 
11  hours,  1  time  30  miles,  and  f  of  another  time,  that  is,  the 
ratio  between  the  distances  will  be  equal  to  the  ratio  be- 
tween the  times. 

-y.  of  30  miles,  that  is,  30  X  V"  ~  H^  =  ^^  miles.  If, 
then,  no  error  has  been  committed,  55  miles  must  be  -y-  of 
30  miles.     This  is  actually  the  case  ;  for  -|3  =  -\K 

>V  Ans,  55  miles. 

Quantities  which  have  the  same  ratio  between  them  are 
said  to  be  praportional.     Thus,  these  four  quantities, 

hours,  hours,     miles,    miles. 

6,  11,  30,  55, 
written  in  this  order,  being  such,  that  the  second  contains 
the  first  as  many  times  as  the  fourth  contains  the  third,  that 
is,  the  ratio  between  the  third  and  fourth  being  equal  to  the 
r^tio  between  the  first  and  second,  form  what  is  called  a  pro- 
portion. ^ 

It  follows,  therefore,  that  proportion  is  a  combinatian  of  two 
i^ual  ratios.  Ralin  exists  between  two  numbers  ;  but  prO' 
porhioii  requires  at  least  three. 


tSO  nULE  OF  THREE.  IT  94, 95. 

To  denote  that  tliere  is  a  proportion  between  the  numbers 
6,  11,  30,  and  55,  they  are  written  thus  : — 

G    !    11    :  :    30    :    55 

which  is  read,  6  is  to  1 1  as  30  is  to  55  ;  that  is,  6  is  the 
same*  part  of  11,  thai  30  is  of  55;  or,  6  is  contained  in  11  as 
many  times  as  30  is  contained  in  55  ;  or,  lastly,  the  ratio  or 
relation  of  11  to  6  is  the  same  as  that  of  55  to  30. 

^  95-  Tlie  first  term  of  a  ratio,  or  relation,  is  called  the 
<mtectdent^  and  the  second  the  consequent.  In  a  proportion 
there  are  two  antecedents,  and  two  conseqnents,  viz.  the  an- 
tecedent of  the  first  ratio,  and  that  of  the  second  ;  ihe  con- 
sequent of  the  first  ratio,  and  that  of  the  second.  In  the 
proportion  6  :  1 1  : :  30  :  55,  the  antecedents  are  6,  30  ;  the 
ccrsequents,  11,  55. 

The  consequent,  as  we  have  already  seen,  is  taken  for  the 
numerator,  and  the  antecedent  for  the  denominator  of  the 
liTiction,  which  expresses  the  ratio  or  relation.  Thus,  the 
lirst  ratio  is  W  the  second  |§-  z=:  Jg^ ;  and  that  these  two 
ratios  are  equal,  we  know,  because  the  fractions  are  equal. 

The  two  fractions  -\^  and  ^J  being  equal,  it  follows  that, 
by  reducing  tliem  to  a  common  denominator,  the  numerator 
of  the  one  wdll  become  equal  to  the  numerator  of  the  other, 
and,  consequently,  that  11  multiplied  by  30  will  give  the 
same  product  as  55  multiplied  by  6.  This  is  actually  the 
case;  for  11  X  30  nz:  330,  and  55  X  6  —  330.  Hence  it 
follows, — If  four  numbers  be  in  proportion^  the  product  of  ihe 
first  and  laM^  or  of  the  two  extremes^  is  equal  to  the  product  of 
the  second  and  thlrd^  or  of  the  two  means. 

Hence  it  will  be  easy,  having  three  terms  in  a  proportion 
ffiven,  to  find  the  fourth.  Take  the  last  example.  Know- 
ing that  the  distances  travelled  are  in  proportion  to  the  times 
or  hours  occupied  in  travelling,  we  write  the  proportion 
thus: — 

hours,    hours.         miles.        mika. 

6    :    11    : :    30 

Now,  since  the  product  of  the  extremes  is  er^ual  to  (he 
product  of  the  means,  w^e  multiply  together  the  two  means, 
11  and  30,  which  makes  330,  and,  dividing  this  product  by 
the  known  extreme,  6,  we  obtain  for  the  result  55,  that  is^ 
W  miles,  which  is  the  other  extreme,  or  term,  sought. 


54    :    186    :  :    9 
9 


IT  95.  HULE    OF    THREE.  181 

3.  At  $  54  for  9  barrels  of  flour,  how  many  barrels  may 
be  purchased  for  $  186  ? 

In  this  question,  the  unknown  quantity  is  the  number  of 
barrels  bought  for  $  186,  which  ought  to  contain  the  9  bar- 
rels as  many  times  as  $  186  contains  $54;  we  thus  get  the 
following  proportion : 

dollars,      dollars.       barrels,    barrels.  The  produCt    1674 

of  the  two  means,  di- 

vided     by     54,     the 

54  )  1674  ( 31  banels,  tk^  Anmer.  known  extreme,  gives 

162  31     barrels     for    the 

other  extreme,  which 

54  is    the    term  sought, 

54  or  Answer, 

Any  three  terms  of  ft  proportion  being  given,  the  operation 
by  which  we  find  the  fourth  is  called  the  Ride  of  lliree,  A 
just  solution  of  the  question  will  sometimes  require,  tiiat  the 
order  of  the  terms  of  a  proportion  be  changed.  This  may 
be  done,  provided  the  terms  be  so  placed,  that  the  product 
of  the  extremes  shall  be  equal  to  that  of  the  means. 

4.  If  3  men  perform  a  certain  piece  of  work  ia  10  days, 
how  long  will  it  take  6  men  to  do  the  same  } 

The  number  of  days  in  which  6  men  will  do  the  work  be- 
ing the  term  sought,  the  known  term  of  the  same  kind,  viz. 
10  days,  is  made  the  third  term.  The  two  remaining  terms 
are  3  men  and  6  men,  the  ratio  of  which  is  f .  But  the 
more*  men  there  are  employed  in  the  work,  the  less  time  will 
be  required  to  do  it;  consequently,  the  da3^s  will  be  less  in 

*  The  nile  of  three  lias  sometimes  been  divided  into  direct  and  i/tre?'?«,  a  dis- 
tinction which  is  totally  useless,  it  may  not  however  be  ajniss  to  explain;  m  this 
place,  in  what  this  distinction  consists. 

The  Rule  of  Three  Direct  is  when  m/)re  rcqn'res  more,  or  less  requires  less,  as 
m  this.example  : — If  3  men  dig  a  trench  48  feet  lon^  in  a  certain  time,  how  many 
feet  will  12  men  di^  in  the  same  time  ?  Here  it  is  obvious,  ilval  [he  rnorp  men 
there  are  employecf,  the  vwre  work  will  be  done  5  and  therefoc-e,  ii:  this  iv.stanoe, 
more  requires  more.  Again : — If  6  men  dig  48  feet  in  a  given  time,  hew  much 
fvill  3  men  di^  in  the  saino  time  ?  Here  less  requires  less,  for  ilie  less  men  there 
are  employefi,  the  less  work  will  he  done. 

The  Ihde  of  Three  Inverse  is  when  more  requires  le^,  or  less  reqi.jires  more,  as 
ill  this  example : — If  G  men  dig  a  certain  quantity  of  trench  in  14  hours,  how  many 
hours  will  it  require  12  men  to  di^  the  same  quantity  ?  Here  more  requirfs  less  ; 
tliat  is,  12  men  beir  g  more  than  6,  wj!J  reqiri'"e  less  time.  Again : — It  i>  men  per- 
form a  piece  of  wor^  in  7  daj'«.  how  long  will  3  men  be  in  pedrnmug  'he  fi».rno 
work  ?  Her«  Us-  requires  viore.  .  i'ov  the  nuniher  tf  men,  \yQiug  less,  wjJi  re<iuiro 
mm:e  time. 

O 


182  RULE    OF    THREE.  IT  95. 

proportion  as  the  number  of  men  is  greater*  There  is  still  a 
proportion  in  this  case,  but  the  order  of  the  terms  ie  inverted; 
for  the  number  of  men  in  the  second  set,  being  two  times 
that  in  the  lirst,  will  require  only  one  half  the  time.  The 
lirst  number  of  days,  therefore,  ought  to  contain  the  second 
as  many  times  as  the  second  number  of  men  contains  the 
first.  This  order  of  the  terms  being  the  reverse  of  that  as- 
signed to  them  in  announcing  the  question,  we  say,  that  the 
number  of  men  is  in  the  inverse  ratio  of  the  number  of  days. 
With  a  view,  therefore,  to  the  just  solution  of  the  question, 
we  reverse  the  order  of  the  two  first  terms,  (in  doing  which 
we  invert  the  ratio,)  and,  instead  of  writing  the  proportion, 
3  men  :  6  men,  (f,)  we  write  it,  6  men  :  3  men,  (f,)  that  is, 

men.    men.        days.         days. 

6    :    3    : :    10    

Note,  We  invert  the  ratio  when  we  reverse  the  order 
of  the  terms  in  the  proportion,  because  then  the  antece- 
dent takes  the  place  of  the  consequent,  and  the  consequent 
that  of  the  antecedent ;  consequently,  the  terms  of  the  frac- 
tion which  express  the  ratio  are  inverted  ;  hence  the  ratio 
is  inverted.  Thus,  the  ratio  expressed  by  §  :=  2,  being  in- 
verted, is  I  m  ^. 

Having  stated  the  proportion  as  above,  we  divide  the  pro- 
duct of  the  means,  (10  X  3  m  30,)  by  the  known  extreme, 
6,  which  gives  5,  that  is,  5  days,  for  the  other  extreme,  or 
term  sought.  Ans.  5  days. 

From  the  examples  and  illustrations  now  given  we  deduce 
the  follov»ing  general 

RULE. 

Of  the  three  given  numbers,  make  that  the  third  term 
which  is  of  the  same  kind  with  the  answer  sought.  Then 
consider,  from  the  nature  of  the  question,  whether  the  an- 
swer will  be  greater  or  less  than  this  term.  If  the  answer  is 
to  be  greater,  place  the  greater  of  the  two  remaining  num- 
bers for  the  second  term,  and  the  less  number  for  the  first 
terra  ;  but  if  it  is  to  be  less,  place  the  less  of  the  two  re- 
maining numbers  for  the  second  term,  and  the  greater  for 
the  first ;  and,  in  either  case,  multiply  the  second  and  third 
terms  together,  and  divide  the  product  by  the  first  for  the 
miswer,  which  will  always  be  of  the  same  denomination  m 
the  third  term. 


It  &5.  RULE    OF    THREE.  18S 

Note  1.  If  the  first  and  second  terms  contain  different  de- 
nominations, they  must  both  be  reduced  to  the  same  de- 
nomination ;  and  if  the  third  term  be  a  compound  number,  it 
either  must  be  reduced  to  integers  of  the  lowest  dtnmninationj 
or  the  low  denominations  must  be  reduced  to  a  fraction  of 
the  highest  denomination  contained  in  it. 

Note  2.  The  same  rule  is  applicable,  whether  the  given 
quantities  be  integral,  fractional,  or  decimal. 

/   '    -^  Examples  for  practice. 

5;  If  6  horses  consume  21  bushels  of  oats  in  3  weeks, 
how  many  bushels  will  serve  20  horses  the  same  time  ? 

Ans,  70  bushels. 

6.  The  above  question  reversed.  If  20  horses  consume  70 
bushels  of  oats  in  3  weeks,  how  many  bushels  will  serve  6 
horses  the  same  time  ?  Ans,  21  bushels. 

7.  If  365  men  consume  75  barrels  of  provisions  in  9 
months,  how  much  will  SCO  men  consume  in  the  same  time  ? 

Ans.  1024|  barrels. 

*  8.  If  500  men  consume  102ff  barrels  of  provisions  in  9 
months,  how  much  will  365  men  consume  in  the  same 
Wme  ?  Ans.  75  barrels. 

9.  A  goldsmith  sold  a  tankard  for  10  i8.  12  s.,  at  the  rate 
of  5  8.  4  d.  per  ounce ;  I  demand  the  weight  of  it. 

Ans.  39  oz.  15  pwt. 

ho.  If  the  moon  move  13°  10'  35^'  in  1  day,  in  what  time 

does  it  perform  one  revolution?        Ans.  27  days,  7  h.  43  m. 

11.  If  a  person,  whose  rent  is  $  145,  pay  $  12^63  parish 
taxes,  how  much  should  a  person  pay  whose  rent  is  $  378  ? 

Ans.  $32^925. 

12.  If  I  buy  7  lbs.  of  sugar  for  75^cents,  how  many  pownds 
can  I  buy  for  $6  ?  '  '     [    "      '  Ans.  56  lbs. 

13.  If  2  lbs.  of  sugar  cost  25  cents,  v/hat  will  100  lbs.  of 
cofi'ee  cost,  if  S  lbs.  of  sugar  are  worth  5  lbs.  of  colfee  ? 

Am.  $20. 
^4.  If  I  give  $6  for  the  use  of  $100  for  12  months, 
what  must  I  give  for  the  use  of  $  357'82  the  same  timp  ? 

Ans.  $21*469. 

^t15.  There  is  a  cistern  which  has  4  pipes;    the  first  will 
fill  it  in  10  minutes,  the  second  in  20  minutes,  the  third  in 


184  RULE    OF    THREE.  IT  95^ 

40  minutes,  and  the  fourth  in  80  minutes;  m  what  time  will 
all  four,  running  together,  fill  it  ^ 

tV  +  rV  +  4^iT  +  A-  ==  J  J  cistern  in  1  minute. 

Alls,  5  J  minutes. 

16.  If  a  family  of  10  persons  spend  3  bushels  of  malt  in 
a  monih,  how  many  bushels  will  serve  them  when  there  are 
30  in  the  family  ?  Am,  9  bushels. 

Note,  The  rule  of  proportion,  although  of  frequent  use, 
is  not  of  indispensable  necessity  ;  for  all  questions  under  it 
may  be  solved  on  general  principles,  without  the  formality 
of  a  propor'*j*on  ;  that  is,  by  analysis^  as  already  shown,  IT  65, 
ex.  1.  Thus",  ill  the  above  example, — If  10  persons  spend 
3  bushels,  1  person,  in  the  same  time,  would  spend  y\j-  of  3 
bushels,  that  is,  -^^^  of  a  bushel ;  and  30  persons  would  spend 
30  times  as  much,  that  is,  -fg-  =.  9  bushels,  as  before. 

■  17.  If  a  staff,  5  ft.  8  in.  in  length,  cast  a  shadow  of  6  feet, 
how  high  is  that  steeple  whose  shadow  measures  153  feet? 

Ans,  144^  feet. 
18.  The  same  by  Gualysls,  If  6  ft.  shadow  require  a  staff 
of  6  ft.  8  in.  nz  68  in.,  1  ft.  shadow  will  require  a  staff  of 
^  of  OS  in.  or  -^^  in. ;  then,  153  ft.  shadow  will  require  153 
times  as  much ;  that  is,  ^^-  X  153  m:  J-0-|G.l  —  1734  in.  = 
144^  ft.,  as  before. 

^^,19.  U  3  £,  sterling  be  equal  to  4  iB.  Massachusetts,  how 

much  Massachusetts  is  equal  to  1000  £,  sterling  ? 

I  Ans,  1333  ie.  6  >.  8  d. 

'  20.  If  1333  <£ .  6  s.  8  d.  Massachusetts,  be  equal  to  1000£ . 

sterling,  how  much  sterling  is  equal  to  4  «£.  Massachusetts? 

Ans,  3  £ , 

21.  If  1000  £  .  sterling  be  equal  to  1333  i3 .  6  s.  8  d.  Mas- 
sachusetts, how  much  Massachusetts  is  equal  to  3  iB.  ster- 
ling ?  Ans,  4  £ , 

22.  If  3  cO  .  sterling  be  equal  to  4  £ .  Massachusetts,  how- 
much  sterling  is  equal  to  1333  £ .  6  s.  8  d.  Massachusetts  ? 

Ans,  1000  ie. 
*  23.  Suppose  2000  soldiers  had  been  supplied  with  bread 
sufficient  to  last:  them  12  weeks,  allowing  each  man  14 
ounces  a  day;  but,  on  examination,  they  find  105  barrels, 
containing  200  lbs.  each,  wholly  spoiled  ;  what  must  the  al- 
lowance be  to  each  man,  that  the  remainder  may  last  them 
the  same  time  ?  Ans,  12  oz.  a  day. 


IT  95.  RULE    OF    THREE.  185 

'  24.  Suppose  2000  soldiers  were  put  to  an  allowance  of 
12  oz.  of  bread  per  day  for  12  weeks,  having  a  seventh  part  of 
their  bread  spoiled ;  what  was  the  whole  weight  of  their 
bread,  good  and  bad,  and  how  much  was  spoiled  ? 

.       J  The  whole  weight,  147000  lbs. 
^^^'  ^  Spoiled,      -       -         21000  fe. 

t25.  2000  soldiers,  having  lost  105  barrels  of  bread, 

weighing  200  lbs.  each,  were  obliged  to  subsist  on  12  oz.  a 
day  for  12  weeks  ;  had  none  been  lost,  they  might  have  had 
14  oz.  a  day  ;  what  was  the  whole  weight,  including  what 
was  lost,  and  how  much  had  they  to  subsist  on  ? 

.       ^  Whole  weight,       147000  lbs. 
^"^*  I  Left,  to  subsist  on,  126000  lbs. 

'  26.  2000  soldiers,  after  losing  one  seventh  part  of 

their  bread,  had  each  12  oz.  a  day  for  12  weeks;  what  was 
the  v/hole  weight  of  their  bread,  including  that  lost,  and  how 
much  might  they  ha\e  had  per  day,  each  man,  if  none  had 
been  lost  ?  C  Whole  weight,  147000  lbs. 

Ans.   <  Loss,      -       -       21000  lbs. 
1  (  14  oz.  per  day,  had  none  been  lost. 

^  27.  There  was  a  certain  building  raised  in  8  months  by 
120  workmen;  but,  the  same  being  demolished,  it  is  required 
to  be  built  in  2  months;  I  demand  how  many  men  must 
be  employed  about  it.  Ans,  480  men. 

28.  There  is  a  cistern  having  a  pipe  which  will  empty  it 
in  10  hours ;  how  many  pipes  of  the  same  capacity  will 
empty  it  in  24  minutes  ?  Ans.  25  pipes. 

29.  A  garrison  of  1200  men  has  provisions  for  9  months, 
at  the  rate  of  14  oz.  per  day ;  how  long  will  the  provisions 
last,  at  the  same  allow^ance,  if  the  garrison  be  reinforced  by 
400  men  ?  Ans.  6 J  months. 

30.  If  a  piece  of  land,  40  rods  in  length  and  4  in  breadth, 
make  an  acre,  how  wide  must  it  be  when  it  is  but  25  rods 
long  ?  Ans.  6f  rods. 

31.  If  a  man  perform  a  journey  in  15  days  when  the  days 
are  12  hours  long,  in  how  many  will  he  do  it  when  the  days 
are  but  10  hours  long  ?  Ans.  18  days. 

32.  If  a  field  will  feed  6  cows  91  days,  how  long  will  it 
feed  21  cows?  A7is.  2G  days. 

33.  Lent  a  friend  292  dollars  for  6  months ;  some  time 
after,  he  lent  me  806  dollars  j  how  long  may  I  keep  it  to 
balance'  the  favour  ?  Aiis.  2  months  6  +  days. 


186  KULE   OF   THREE.  t  &5. 

34.  If  30  men  can  perform  apiece  of  work  in  11  days, 
how  many  men  will  accomplish  another  piece  of  work,  4 
times  as  big,  in  a  fifth  part  of  the  time  ?  Am,  600  men. 

35.  If  -J^  lb.  of  sugar  cost  f  ^  of  a  shilling,  what  will  ^f 
of  a  lb.  cost  ?  Am,  4  d.  3^g  J-^  q. 

Note,  See  IT  65,  ex.  1,  where  the  above  question  is 
solved  by  analysis.  The  eleven  following  are  the  next  suc- 
ceeding examples  in  the  same  IT. 

I  ?6.  If  7  lbs.  of  sugar  cost  f  of  a  dollar,  what  cost  12  lbs.  ? 

Am,  $lf 
^  37.  If  64  yds.  of  cloth  cost  $3,  what  cost  9^  yds.  ? 

Am,  $4*269. 
's  38.  If  2  oz.  of  silver  cost  $  2*24,  what  costs  f  oz.  ? 

Am.  $0*84. 

39.  Iff  oz.  cost  $|J-,  what  costs  1  oz.  ?       Am,  $  1*283. 

40.  If  4  lb.  less  by  ^  lb.  cost  13^  d.,  what  cost  14  lbs. 
less  by  -J  of  2  lbs.  ?  Ans.  4  £ .  9  s.  9^^  d. 

41.  If  f  yd.  cost  $  ^,  what  will  40^  yds.  cost  ? 

Am,  $59*062. 
f  42*  If  /^  of  a  ship  cost  $  251,  what  is  ^^g-  ^^  ^^^^  worth  r 
^  Am.  $53*785. 

43.  At  3|  £ ,  per  cwt.,  what  will  9f  lbs.  cost  ? 

Am,  6  s.  3-5^^  d. 

44.  A  merchant,  OAvning  ^  of  a  vessel,  sold  f  of  his  share 
for  $  957  ;  what  was  the  vessel  worth  ?      Aiis.  $  1794*375. 

45.  If  I  yd.  cost  f  ^ .,  what  will  y\  of  an  ell  English  cost  ? 

A71S,  17  s.  1  d.  2f    q. 

46.  A  merchant  bought  a  number  of  bales  of  velvet,  each 
containing  129^-f  y<?s.,  at  the  rate  of  $  7  for  5  yds.,  and  sold 
them  out  at  the  rate  of  $  11  for  7  yds.,  and  gained  $200 
by  the  bargam  ;  how  many  bales  were  there  ?    Am.  9  bales. 

47.  At  $  33  for  6  barrels  of  flour,  what  must  be  paid  for 
178  barrels?  Am.  $979. 

48.  At  $  2*25  for  3*17  cwt.  of  hay,  how  much  is  that  per 
ton?  Am.  $14*196. 
\  49.  If  2*5  lbs.  of  tobacco  cost  75  cents,  how  much  will 
185  lbs.  cost  ?  Am,  $  5*55. 

50.  What  is  the  value  of  *  15  of  a  hogshead  of  lime,  at 
$  2*39  per  hhd.  ?  Am.  $  0*3585. 

61.  If  *15  of  a  hhd.  of  lime  cost  $  0*3585,  what  is  it  pet 
litid.?  Ans.  $2*39. 


IT  06.  COMPOUND    PROPORTION.  187 


COMPOUND  PROPORTION. 

Vi  96.  It  frequ  -ntly  happens,  that  the  relation  of  the 
quantity  required,  to  the  given  quantity  of  the  same  kind, 
depends  upon  several  circumstances  combined  together;  it 
is  then  called  Coynpound  Proportion^  or  Double  Rule  of  Three, 

1.  If  a  man  trdvel  273  miles  in  13  days,  travelling  only 
7  hours  in  a  day,  how  many  miles  will  he  travel  in  12  days, 
if  he  travel  10  hours  in  a  day  ? 

This  question  may  be  solved  several  ways.  First,  by  analy- 
sis:— 

If  wc  knew  how  many  miles  the  man  travelled  in  1  hour, 
it  is  plain,  we  might  take  this  number  10  times,  which  would 
be  the  number  of  miles  he  would  travel  in  10  hourg,  or  in  1 
of  these  long  days,  and  this  again,  taken  12  times,  would  be 
the  number  of  miles  he  would  travel  in  12  davs.  travelling 
10  hours  each  day. 

^  If  he  travel  273  miles  in  13  days,  he  will  travel  y^  of  273 
miles  ;  that  is,  ^■^-  miles  in  1  day  of  7  hours ;  and  f  of  ^/t^ 
miles  is  ^^^-  miles,  the  distance  he  travels  in  1  hour :  then, 
10  times  -^^-^  1=  ^-pp-  nriles,  the  distance  he  travels  in  10 
hours;  and  12  times  ^^.fo  —  32^_^6ii  —  350  miles,  the  dis- 
tance he  travels  in  12  days,  travelihig  10  hours  each  day. 

Ans.  360  miles. 

But  the  object  is  to  show  how  the  question  may  be  solved 
by  proportion : — 

First;  it  is  to  be  regarded,  that  the  number  of  miles  tra- 
veiled  over  depends  upon  two  circumstances,  viz.  the  num- 
ber of  days  the  man  travels,  and  the  number  of  hours  he 
travels  each  day. 

We  will  not  at  first  consider  this  tatter  circumstance,  but 

■  suppose  the  number  of  hours  to  be  the  same  in  each  case  : 

the  question  then  will  be, — If  a  man  travel  273  miles  in  13 

days^  how  many  miles  will  he  travel  in  12  days?     This  will 

furnish  the  following  proportion  : — 

13  days    :    12  days    :  :    273  miles    :    miles 

which  gives  for  the  fourth  term,  or  answer,  252  miles. 

Now,  taking  into  consideration  the  other  circumstance,  of 
that  of  the  hours,  we  must  say, — If  a  man,  travelling  7  hours 
«  day  for  a  certain  number  of  daysy  travels  252  mUeSy  how  fat 


188  COMPOUND    PROPORTION.  IT  96. 

ioill  he  travel  in  the  same  time^  if  he  travel  10  hours  in  a  day  ? 
This  will  lead  to  the  following  proportion  : — 

7  hours    :    10  hours    :  :    252  miles    :    miles. 

This  gives  for  the  fourth  term,  or  answer,  360  miles. 

We  see,  then,  that  273  miles  has  to  the  fourth  term,  or 
answer,  the  same  proportion  that  13  days  has  to  12  days, 
and  that  7  hours  has  to  10  hours.  Stating  this  in  the  form 
of  a  proportion,  we  have 

13  days      :    12  days   J    ..    273  miles   :    miles 

7  hours    :    10  hours  ^ 

by  which  it  appears,  that  273  is  to  be  multiplied  by  both  12 
and  10  ;  that  is,  273  is  to  be  multiplied  by  the  product  of 
12  X  10,  and  divided  by  the  product  of  13'  X  7,  which,  be- 
ing done,  gives  360  miles  for  the  fourth  term,  or  answer,  as 
before. 

In  the  same  manner,  any  question  relating  to  compound 
proportion,  however  complicated,  may  be  stated  and  solved. 

^  2.  If  248  men,  in  5  days,  of  1 1  hours  each,  can  dig  a  trench 
230  yards  long,  3  wide,  and  2  deep,  in  how  many  days,  of  9 
hours  each,  will  24  men  dig  a  trench  420  yards  long,  5  wide, 
and  3  deep  ? 

Here  the  number  of  days,  in  which  the  proposed  work  can 
be  done,  depends  on  Jive  circumstances^  viz.  the  number  of 
men  employed,  the  number  of  hours  they  work  each  day, 
the  length,  breadth,  and  depth  of  the  trench.  We  will  con- 
sider the  question  in  relation  to  each  of  these  circumstances, 
in  the  order  in  which  they  have  been  named : — 

1st.  The  number  of  men  employed.  Were  all  the  circum- 
stances in  the  two  cases  alike,  except  the  number  of  men  and 
the  number  of  days,  the  question  would  consist  only  in  find- 
ing in  how  many  days  24  men  would  perform  the  work  which 
248  men  had  done  in  5  days ;  we  should  then  have 

24  men    :    248  men    :  :    5  days    :    days. 

2d.  Hours  in  a  day.  But  the  first  labourers  worked  11 
hours  in  a  day,  whereas  the  others  worked  only  9 ;  less  hours 
will  require  more  days,  which  will  give 

9  hours    :    11  hours    ;:    5  days    :    days. 

3d.  Length  of  the  ditches.     The  ditches  being  of  UDcqiml 


V  96,  97.  COMPOUND    PROPORTlOjy.  189 

length,  as  many  more  days  will  be  necessary  as  the  second 
is  longer  than  the  first;  hence  we  shall  have 

230  length    :    420  length   :  :    5  days    :    days. 

4th.    Widlhs.    Taking  into  consideration  the  widths,  which 
are  different,  we  have 

3  wide    :    5  wide   :  :    5  days    :    days. 

5th.  .Depths.     Lastiy,  the  depths  being  different,  we  have 

2  deep    :    3  deep    :  :    5  days   :    days. 

It  would  seem,  therefore,  that  5  days  has  to  the  fourth 
term,  or  answer,  the  same  proportion 

tkat   24  men     has  to  248  men,         whose   ratio   is      ^^, 
that      9  hours  has  to    11  hours,  the  ratio  of  which  is  \^, 

that  230  length  has  to  420  length, f|§, 

that      3  width  has  to      5  width, f, 

that     2  depth  has  to      3  depth,   f; 

all  which  stated  in  form  o[  a  proportion,  we  have 


davs. 


IT  97.  The  continued  product  of  all  the  second  terms 
248  X  11  X  420  X  5  X  3,  multiplied  by  the  third  term, 
6  days,  and  this  product  divided  by  the  continued  ]^ro- 
duct  of  the  first  terms,  24  X  9  X  230  X  3  X  2,  gives 
288^3^\9^G^^j  clays  for  the  fourth  term,  or  answer.     2882%^. 

Bu^t  the  first  and  second  terms  are  the  fractions  ^^V^,  V, 
jl^.^  ^  and  f,  which  express  the  ratios  of  the  men,  and  of 
the  hours,  of  the  lengths,  widths  and  depths  of  the  two 
ditches.  Hence  it  follows,  that  the  ratio  of  the  number  of 
days  given  to  the  number  of  days  sought,  is  equal  to  the  pro- 
duct of  all  the  ratios,  which  result  from  a  comparison  ot  the 
terms  relating  to  each  circumstance  of  the  question. 

The  product  of  all  the  ratios  is  found  by  multiplying  to- 

,  .,        248  X  llX  420 

gether  the  fractions  which  express  them,  thus,  2i~x  y  X  230 

^gl  =  ^^'  -<1  this  fraction,  H— ,  represents  tl.e 


Men, 

24 

2481 

Hours, 

9 

11 

common  tonn. 

Length, 

230 

420 

^     :  :     5  days 

Width, 

3 

0 

Depth, 

2 

3. 

iSO  COMPOUND   PROrORTION.  1}  Otv 

ratio  of  the  quantity  required  to  the  given  quantity  of  the  same 
kind.  A  ratio  resulting  in  this  manner,  from  the  multiplica- 
tion of  several  ratios,  is  called  a  compound  ratio. 

From  the  examples  and  illustrations  now  given  we  de« 
duce  the  following  general 

RULE 

for  solving  questions  in  compound  proportion,  or  double 
rule  of  three,  viz. — Make  that  number  which  is  of  the 
same  kind  with  the  required  answer,  the  third  term ;  f^ad, 
of  the  remaining  numbers,  take  away  two  that  are  of  the 
same  kind^  and  arrange  them  according  to  the  directions 
given  in  simple  proportion ;  then,  any  other  two  of  the  sajae 
kind,  and  so  on  till  all  are  used. 

Lastly,  multiply  the  third  term  by  the  continued  product 
of  the  second  terms,  and  divide  the  result  by  the  continued 
product  of  the  first  terms,  and  the  quotient  will  be  the  fourth 
term,  or  answer  required. 

EXAMPLiES    FOR    PRACTICE. 

1.  If  6  men  build  a  wall  20  ft.  long,  6  ft.  high,  and  4  ft 
thick,  in  16  days,  in  what  time  will  24  men  build  one  200 
ft.  long,  8  ft.  high,  and  6  ft.  thick  ?  A7is.  80  days. 

2.  If  the  freight  of  9  hhds.  of  sugar,  each  weighing  12 
cwt.,  20  leagues,  cost  16  £> .,  what  must  be  paid  for  the 
freight  of  50  tierces,  each  weighing  2^  cwt.,  100  leagues  ? 

Ans.  92  je.  11  s.  lOf  d. 

8.  If  56  lbs.  of  bread  be  sufficient  for  7  men  14  days,  how 
much  bread  will  serve  21  men  3  days  ?  Ans.  30  lbs. 

The  same  by  analysis.  If  7  men  consume  56  lbs.  of  bread, 
1  man,  in  the  same  time,  would  consume  4-  of  56  lbs.  =: 
^-  lbs. ;  and  if  he  consume  -^^  lbs.  in  14  days,  he  would 
consume  y^  o(  Af-  =:  ||  lb.  in  1  day.  21  men  would  con- 
sume 21  times  so  much  as  1  man;  that  is,  21  times  f||  1= 
J-|-P-  lbs.  in  1  day,  and  in  3  days  they  would  consume  3 
times  as  much ;  that  is,  ^  |-^  =  36  lbs.,  as  before. 

Ans.  36  lbs. 

Note.  Having  wrought  the  following  examples  by  the 
rule  of  proportion,  let  the  pupil  be  required  to  do  the  same 
by  analysis. 

4.  If  4  reapers  receive  $11*04  for  3  days'  work,  how 
many  men  may  be  hired  16  days  for  $  103'04  ? 

Am.  7  men. 


'tr  97.    SUPPLEMENT  TO  THE  SINGLE  RULE  OF  THREE.    191 

5.  If  7  oz.  5  pwt.  of  bread  be  bought  for  4|  d.  when  com 
is  4  s.  2  d.  per  bushel,  what  weight  of  it  may  be  bought  for 
1  s.  2  d.  when  the  price  per  bushel  is  5  s.  6  d.  ? 

Ans,  1  lb.  4  oz.  3|Jf  pwts- 

6.  If  $100  gain  $6  in  1  year,  what  will  $400  gain  in 
9  months  ? 

Note.  This  and  the  three  following  examples  reciprocally 
prove  each  other. 

7.  If  $  100  gain  $6  in  1  year,  in  what  time  will  $400 
gain  $  18  ? 

8.  If  $  400  gain  $  18  in  9  months,  what  is  the  rate  per 
cent,  per  annum  ? 

\9.  What  principal,  at  6  percent,  per.  ann.,  will  gain  $  18 
in  9  months  ? 
-  "i     10.  A  usurer  put  out  $75  at  interest,  and,  at  the  end  of  8 
^    months,  received,  for  principal  and  interest,  $  79 ;  I  demand 
ftt  what  rate  per  cent,  he  received  interest. 

! '  /  Ans.  8  per  cent. 

11.  If  3  mei^^rficeive  fe^^  jB .  for  19 J- days' work,  how 
much  must  20  men  receive  for  100^  days'  ? 

A71S.  305  je.  Os.  8d, 


SXTPPIJGIiyCSNT   TO  THE   SZnC^Z.£S   HUXiB    OP 
TBUXIS. 

QUESTIONS. 

1.  What  is  proportion?     2.  How  many  numbers  are  re- 
quired to  form  a  ratio  ?     3.  How  many  to  form  a  proporticfn  ? 

4.  What  is  the  first  term  of  a  ratio  called  ?  5.  the  second 

term  ?  6 .  Which  is  taken  for  the  numerator,  and  which  for 
the  denominator  of  the  fraction  expressing  the  ratio  ?  7. 
How  may  it  be  known  when  four  numbers  are  in  proportion  ? 
8.  Having  three  terms  in  a  proportion  given,  how  may  the 
fourth  term  be  found  ?  9.  What  is  the  operation,  by  which 
the  fourth  term  is  found,  called  ?  10.  How  does  a  ratio  be- 
come inverted  ?  11.  What  is  the  rule  in  proportion?  12. 
In  what  denomination  will  the  fourth  term,  or  answer,  be 
found?  13.  If  the  first  and  second  terms  contain  different 
denominations,  what  is  to  be  done  ?  14.  What  is  cqmpound 
proportion,  or  double  rule  of  three  ?     15.  Rule  ? 


193  FELLOWSHIP.  IT  97,  98, 

EXERCISES. 

1.  If  I  buy  76  yds.  of  cloth  for  $11347,  what  does  it 
cost  per  ell  English  ?  Ans.    $  1^861, 

2.  Bought  4  ]iieces  of  Holland,  each  containing  ^4  eils 
English,  for  $  96  ;  how  much  was  that  per  yard  ? 

Ans.    $0*80. 

3.  A  garrison  had  provision  for  8  months,  at  the  rate  ol 
15  ounces  to  each  person  per  day  ;  how  much  must  be  al 
lowed   per   day,  in  order  that   the  provision  may  last  9^ 

Months?  Ans.   12ff  oz.^ 

4.  How  much  land,  at  $  2'50  per  acre,  must  be  given  in 
exchange  for  360  acres,  at  $  3'75  per  acre  ? 

Am,  540  acres, 

5.  Borrowed  185  quarters  of  corn  when  the  price  was 
19  s. ;  how  much  must  I  pay  when  the  price  is  17  s.  4  d.  ? 

Ans.  202f4-. 

6.  A  person,  owning  -g  of  a  coal  mine,  sells  f  of  his  share 
for  171^. ;  what  is  the  whole  mine  worth  ?       Ans.  380^2. 

7.  If  |-  of  a  gallon  cost  f  of  a  dollar,  what  costs  |  of  a 
tun?  Ans.    $140. 

8.  At  1^  iS .  per  cwt,  what  cost  3^  lbs.  ?  Ans.  lOJ-  d. 

9.  If  4-j-  cwt.  can  be  carried  36  miles  for  35  shillings,  how 
many  pounds  can  be  carried  20  miles  for  the  same  money  ? 

Ans.  907^  Ibg. 

10.  If  the  sun  appears  to  move  from  east  to  west  360  de- 
crees in  24  hours,  hov/  much  is  that  in  each  hour  ?  in 

each  minute  ?  in  each  second  ? 

A'/is.  to  last,  15"  of  a  deg. 

11.  If  a  family  of  9  persons  spend  $  450  in  5  months,  how 
much  would  be  sufficient  to  maintain  them  8  months  if  5 
persons  more  were  added  to  the  family?  Ans.  $  1120. 

Note.  Exercises  14th,  15th,  16th,  17th,  18th,  19th,  and 
*?Oth,  "  Supplement  to  Fractions,^^  afford  additional  example* 
in  single  and  double  proportion,  should  more  examples  be 
thought  necessary. 


FSZiKOWSlIIP. 

If  98.  1.  Two  men  own  a  ticket;  the  first  owns  ^,  and 
the  second  owns  f  of  it ;  the  ticket  draws  a  prize  of  40  del- 
ars ;  what  is  each  man's  share  of  the  money  ? 


K  98.  FELLOWSHIP  ISS 

2»  Two  men  purchase  a  ticket  for  4  dollars,  of  which  one 
pays  1  dollar,  and  the  other  3  dollars ;  the  ticket  draws  4^ 
<iollars ;  what  is  each  man's  share  of  the  money  ? 

3.  A  and  B  bought  a  quantity  of  cotton ;  A  ]»aid  100 
dollars,  and  B  200  dollars;  they  sold  it  so  as  to  gain  30 
dollars ;  what  were  their  respective  shares  of  the  gain  ? 

The  process  of  ascertaining  the  respective  gains  or  losses 
of  individuals,  engaged  in  joint  trade,  is  called  the  Rule  of 
Fellowsfiip. 

The  money,  or  value  of  the  articles  employed  in  trade,  is 
called  the  Capitol^  or  Stock  ;  the  gain  or  loss  to  be  shared  is 
called  the  Dividend, 

It  is  plain,  that  each  maii^s  gain  or  loss  ought  to  have  the 
same  relation  to  the  wiiole  gain  or  loss,  as  his  share  of  the 
stock  does  to  the  whole  stock 

Hence  we  have  this  Rule  :— As  the  whole  stock :  to  each 
man's  share  of  the  stock  :  :  the  whole  gain  or  loss  :  his  share 
iof  the  gain  or  loss. 

4.  Two  persons  have  a  joint  stock  in  trade ;  A  put  in 
$  250,  and  B  $  350  ;  they  gain  $  400 ;  what  is  each  man's 

share  of  the  profit? 

OPERATlOxN. 
A's  stock,      $250  \  Then, 

B's,  stock,  $350  f  QQQ  :  250  :  :  400  :  166'G60§  dolls.  A!s  gain. 
Whole  stock,  $G00  )  GOO  ;  350  :  :  400  :  2a3'333J  dolla.  B's  gain. 

The  pupil  will  perceive,  that  the  process  may  be  contract- 
ed by  cutting  off  an  equal  number  of  ciphers  from  the  fffst 
and  secondj  or  first  and  third  terms ;  thus,  6  :  250  :  :  4  : 
166^666f,  &c. 

It  is  obvious,  the  correctness  of  thie  work  may  be  ascer- 
tained by  finding  whether  the  sums  of  the  ^toes  of  the  gains 
are  equal  to  the  whole  gain ;  thus,  $  l^«666f  +  $233^333^ 
=  $  400,  whole  gain. 

^  5.  A,  B  and  C  trade  in  company ;  A?4i  capital  was  $  175, 
B's  $200,  andC's  $500;  by  misfortune  they  lose  $250; 
what  loss  must  each  sustain  ?  C  $  ^0',        A'a  loss. 

Am.  {  $  5n4Sf,  B's  loss. 
(  $142<a57J,C'slo^s. 

6.  Divide  $600  among  3  persons,  so  that  their  shardU 
may  be  to  each  other  as  1 ,  2,  3,  respectively. 

An3.   $10Q,  $200,  and  |^300, 


194  FI5LLOWSH1P.  IT  95 

7.  Two  merchants,  A  and  B,  loaded  a  ship  with  600 
hhds.  of  rum ;  A  loaded  350  hlids.,  and  B  the  rest ;  in  a 
storm,  the  seamen  were  obliged  to  throw  overboard  100 
hhds. ;  how  much  must  each  sustain  of  the  loss  ? 

Am,  A  70,  and  B  30  hhds. 

8.  A  and  B  companied ;  A  put  in  $  45,  and  took  out  f 
of  the  gain ;  how  much  did  B  put  in  ?  Ans.    $  30. 

Note.  They  took  out  in  the  same  proportion  as  they  put 
in ;  if  3  fifths  of  the  stock  is  $  45,  how  much  is  2  fifths 
of  it  ? 

f9.  A  and  B  companied,  and  trade  with  a  joint  capital  of 
$  400 ;  A  receives,  for  his  share  of  the  gain,  ^  as  much  as  B ; 
what  was  the  stock  of  each  ? 

.       {  $133^333^,  A's  stock. 

^"**  I  $  266'666f ,  B's  stock. 

10.  A  bankrupt  is  indebted  to  A  $  780,  to  B"  $  460,  and 

to  C  $  760 ;  his  estate  is  worth  only  $  600 ;  how  must  it 

be  divided  ? 

Note.  The  question  evidently  involves  the  principles  of 
fellowship,  and  may  be  wrought  by  it. 

Ans.  A  $234,  B  $  138,  and  C  $228. 

'll.  A  and  B  venture  equal  stocks  in  trade,  and  clear 

$  164 ;    by  agreement,  A  was  to  have  5  per  cent,  of  the 

j>rofits,  because  he  managed  the  concerns ;  B  was  to  have 

but  2  per  cent. ;  what  was  each  one's  gain  ?  and  how  much 

did  A  receive  for  his  trouble  ? 

Ans.  A's  gain  was  $117'142f,  and  B's  $46,857|,  and 
A  received  $  70'285f  for  his  trouble. 

\  12.  A  cotton  factory,  valued  at  $  12000,  is  divided  into 
100  shares ;  if  the  profits  amount  to  15  per  cent,  yearly,  what 

will  be  the  profit  accruing  to   1   share  ? to  2  shares  ? 

to  5  shares  ?  to  25  shares  ? 

Ans.  to  the  Just.   $  450. 

13.  In  the  above-mentioned  factory,  repairs  are  to  be  made 

which  will  cost  $  340  ;  what  will  be  the  tax,  on  each  share, 

necessary  to  raise  the  sum  ?  on  2  shares  ?  on  3 

shares  ?  on  10  shares  ?  Ans.  to  the  last^   $34. 

i     14.  If  a  town  raise  a  tax  of  $  1850,  and  the  whole  town 

//be  valued  at  $37000,  what  will  that  be  on   $1?  What 

will  be  the  tax  of  a  man  whose  property  is  valued  at  $  1780  ? 

Ans.   $*05  on  a  dollar,  and  $89  on  $1780. 


ir99- 


FELLOWSHIP. 


195 


IT  M.  In  assessing  taxes,  it  is  necessary  to  have  an  In- 
tentory  of  the  property,  both  real  and  personal,  of  the  whole 
town,  and  also  of  the  whole  number  of  polls ;  and,  as  the  polls 
are  rated  at  so  much  each,  we  must  first  take  out  from  the 
whole  tax  what  the  polls  amount  to,  and  the  remainder  is  to 
be  assessed  on  the  property.  We  may  then  find  the  tax  upon 
1  dollar,  and  make  a  table  containing  the  taxes  on  1,2,  3, 
&c.,  to  10  dollars ;  then  on  20,  30,  &.C.,  to  100  dollars ;  and 
then  on  100,  200,  &:c.,  to  1000  dollars.  Then,  knowing  the 
inventory  of  any  individual,  it  is  easy  to  find  the  tax  upon  his 
property. 

15.  A  certain  town,  valued  at  $64,530,  raises  a  tax  of 
$2259*90;  there  are  540  polls,  which  are  taxed  $*60 
each  ;  what  is  the  tax  on  a  dollar,  and  what  will  be  A's  tax, 
xvhose  real  estate  is  valued  at  $  1340,  his  personal  property 
at  $  874,  and  who  pays  for  2  polls  ? 

540  X  *60  z=  $324,  amount  of  the  poll  taxes,  and 
$2259*90  —  $324  =  1935*90,  to  be  assessed  on  property. 
$64530  :  $1935*90  : :  §1  :  *03;  or,  x|f|4^=:*03,taxon  $1. 


TABLE. 


dolls.      dollR. 

Tax  on  1  is  *03 


dollfl.  dolls.  dolls. 

Tax  on  10  is    *30   Tax  on  100 

2  ..   *06    20  ..     *60    200 

3  ..   *09    30  ..     *90   300 

4..    *12    40  ..  1*20    400 

5  ..  *15    50  ..  1*50   500 

6  ..  *18    60  ..  1*80   600 

7  ..  *21     70  ..  2*10    700 

8  ..  *24    80  ..  2*40    800 

9  ..  *27    90  ..  2*70    900 

1000 

Now,  to  find  A's  tax,  his  real  estate  being  $  1340, 
by  the  table,  that 

The  tax  on     -     -    -     $1000     -    -    is    -  -     j 

The  tax  on      -     -     -           300     .    -     -     •  - 

The  tax  on     -    -    -            40    -    -     -     -  - 

Tax  on  his  real  estate   -----     -     -- 

In  like  manner  I  find  the  tax  on  his  personal  > 
property  to  be        ....>....^ 

2  polls  at  ^0  each,  are        .....•• 


doDs. 

is  3* 
..  6* 
..  9^ 
..  12* 
..  15* 
..  18* 
..  21* 
..  24* 
..  27* 
..  30* 
I  find, 

^30* 
9* 
1*20 


Amoyntj  $67*62 


la^  FELLOWSMP.  IT  99,  100, 

Vl6.  What  will  B's  tax  amount  to,  whose  inventory  is  874 
dollars  reaL  and  210  dollars  perscmal  property,  and  who  pays 
%  3  polls  ?  '  Am,    $  34'32l 

17.  What  will  be  the  tax  of  a  man,  paying  for  1  poll, 

whose  property  is  valued  at    $  3482  ?  at  $  768  ?  - 

at  $  940  ?  at  $  4657  ?  Am,  to  the  last^    $  140^31. 

18.  Two  men  paid  10  dollars  for  the  use  of  a  pasture  1 
month ;  A  kept  in  24  cows,  and  B  16  cows ;  how  much 
should  each  pay  ? 

19.  Two  men  hired  a  pasture  for  $  10  ;  A  put  in  8  cows 
3  months,  and  B  put  in  4  cows  4  months ;  how  much  should 
each  pay  ? 

IT  10(^.  The  pasturage  of  8  cows  for  3  months  is  the 
same  as  of  24  cows  for  1  month,  and  the  pasturage  of  4  cows 
for  4  months  is  the  same  as  of  16  cows  for  1  month.  The 
shares  of  A  and  B,  therefore,  are  24  to  16,  as  in  the  former 
question.  Hence,  when  time  is  regarded  in  fellowship, — 
Multiply  each  one'^s  stock  by  the  time  he  continues  it  in  tradey 
and  use  the  product  for  his  share.  This  is  called  Double  Fel-* 
lotvship,  Ans.  A  6  dollars,  and  B  4  dollars. 

n20.  a  and  B  enter  into  partnership  ;  A  puts  in  $  100 
0  months,  and  then  puts  in  $  50  more ;  B  puts  in  $  200  4 
months,  and  then  takes  out  $  80  ^  at  the  close  of  the  year, 
tliey  find  that  they  have  gained  $  95  ;  what  is  the  profit  of 
each  ?  .  A       ^  $  43'711,  A's  share. 

^"^*  I  $51^288,  B's  share. 
Ik21.  A,  with  a  capital  of  $500,  began  trade  Jan.  1,  1826, 
and,  meeting  with  success,  took  in  B  as  a  partnery  with  a_. 
capital  of  $  600,  on  the  first  of  March  following ;  four 
months  after,  they  admit  C  as  a  partner,  who  brought  $  800 
stock ;  at  the  close  of  the  year,  they  find  the  gain  to  be 
$  700 ;  how  must  it  be  divided  among  the  partners  ? 

C  $.  250,  A's  share. 

Am.  <  $  250,  B's  share. 

(  $  200,  C's  share. 

QUESTIONS, 

1.  What  is  fellowship  ?  2.  What  is  the  rule  for  operat- 
ing? 3.  When  time  is  regarded  in  fellowship,  what  is  it 
called?  4.  What  is  the  method  of  operating  in  double 
fellowship  ?  6.  How  are  taxes  assessed .?  6,  How  is 
feilowship  proved  ? 


fl  101,  102.  ALLIGATION.  197 


IT  101.  Alligation  is  the  method  of  mixing  two  or  more 
gimples,  of  different  qualities,  so  that  the  composition  may  be 
of  a  mean,  or  middle  quality. 

When  the  quantities  and  prices  of  the  simples  are  given, 
to  find  the  mean  price  of  the  mixture,  compounded  of  them, 
the  process  is  called  Alligation  Medial, 

1.  A  farmer  mixed  together  4  bushels  of  wheat,  worth 
150  cents  per  bushel,  3  bushels  of  rye,  worth  70  cents  per 
bushel,  and  2  bushels  of  corn,  worth  50  cents  per  bushel; 
what  is  a  bushel  of  the  mixture  worth  ? 

It  is  plain,  that  the  cost  of  the  whole^  divided  by  the  ntm^ 
her  of  bushels,  will  give  the  price  of  one  bushel, 

4  bushels,  at  150  cents,  cost  600  cents. 

3  at    70  210 

2  at    50  100         ^^  =1  lOli  cts,  Ans, 

9  bushels  cost  910  cents. 

2.  A  grocer  mixed  5  lbs.  of  sugar,  worth  10  cents  per  lb., 
8  lbs.  worth  12  cents,  20  lbs.  worth  14  cents ;  what  is  a 
pound  of  the  mixture  worth  ?  Am,   12^  J. 

3.  A  goldsmith  melted  together  3  ounces  of  gold  20 
carats  fine,  and  5  ounces  22  carats  fine ;  what  is  the  fine- 
ness of  the  mixture  ?  Ans,  21|^. 

4.  A  grocer  puts  6  gallons  of  water  into  a  cask  containing 
40  gallons  of  rum,  worth  42  cents  per  gallon ;  what  is  a  gal- 
lon of  the  mixture  woith  ?  Ans,  36 J§  cents. 

5.  On  a  certain  day  the  mercury  was  observed  to  stand  in 
the  thermometer  as  follows  :  6  hours  of  the  day,  it  stood  at 
64  degrees ;  4  hours,  at  70  degrees ;  2  hours,  at  75  degrees, 
and  3  hours,  at  73  degrees:  what  was  the  mean  temperature 
for  that  day  ? 

It  is  plain  this  question  does  not  differ,  in  the  mode  of  its 
operation,  from  the  former.  Ans,  69-^^  degrees. 

IT  102.  When  the  mean  price  or  rate,  and  the  prices  or 
rates  of  the  several  simples  are  given,  to  find  the  proportion$ 
or  quantities  of  each  simple,  the  process  is  called  ALrgatim 
Alternate :  alligation  alternate  is,  therefore,  the  reverse  of 
alligatiou  medial,  and  may  be  proved  by  it 


19S  Ai.i.i«ATio]sr.  t  102v 

1.  A  man  has  oats  worth  40  cents  per  bushel,  which  he 
wishes  to  mix  with  corn  worth  50  cents  per  bushel,  so  that 
the  mixture  may  be  worth  42  cents  per  bushel ;  w^hat  pro- 
portions, or  quantities  of  each,  must  he  take  ? 

Had  the  price  of  the  mixture  required  exceeded  the  price 
of  the  oats,  by  just  as  much  as  it  fell  short  of  the  price  of 
the  corn,  it  is  plain,  he  must  have  taken  equal  quanlilies  of 
oats  and  corn ;  had  the  prince  of  the  mixture  exceeded  the 
price  of  the  oats  by  only  J  as  much  as  it  fell  short  of 
the  price  of  the  corn,  the  compound  would  have  required  2 
times  as  much  oats  as  corn  ;  and  in  all  cases,  the  less  the 
difference  between  the  price  of  the  mixture  and  that  of  one 
of  the  simples,  the  greater  must  be  the  quantity  of  that  sim- 
phj  in  proportion  to  the  other  ;  that  is,  the  quantities  of  the 
simples  must  be  wverseli/  as  the  differences  of  their  prices 
from  the  price  of  tlie  mixture ;  therefore,  if  these  differen- 
ces be  mutually  excha7iged,  they  will,  directly^  express  the 
relative  quantities  of  each  simple  necessary  to  form  the  com- 
pound required.  In  the  above  example,  the  price  of  the 
mixture  is  42  cents,  and  the  price  of  the  oats  is  40  cents ; 
consequently,  the  difference  of  their  prices  is  2  cents  :  the 
price  of  the  corn  is  50  cents,  which  dificrs  from  the  price 
of  the  mixture  by  8  cents.  Therefore,  by  exchanging  these 
differencee,  we  have  8  bushels  of  oats  to  2  bushels  of  corn^ 
for  the  proportion  required. 

Alts.  8  bushels  of  oats  to  2  bushels  of  corn^  or  in  that 
proportion. 

The  correctness  of  this  result  may  now  be  ascertained  by 
the  last  rule ;  thus,  the  cost  of  8  bushels  of  oats,  at  40  cents, 
is  320  cents;  and  2  bushels  of  corn,  at  50  cents,  is  100 
cents ;  then,  320  -f- 100  z=:  420,  and  420,  divided  by  the  num- 
ber of  bushels,  (8  -}-  2,)  =  10,  gives  42  cents  for  the  price  of 
the  mixture. 

2.  A  merchant  has  several  kinds  of  tea ;  some  at  8  shil- 
lings, some  at  9  shillings,  some  at  11  shillings,  and  some 
at  12  shillings  per  pound ;  what  proportions  of  each  must 
lie  mix,  that  he  may  sell  the  compound  at  10  shillings  per 
pound  ? 

Here  we  have  4  simples;  but  it  is  plain,  that  what  has 
just  been  proved  of  two  will  apply  to  any  number  of  pairs, 
if  in  each  pair  the  price  of  owe  simple  is  greater,  and  that  of 
the  otlier  less,  than  the  price  of  the  mixture  required. 
Hence  we  have  this 


T103,  ALLIQATIOI?.  1^9 


RULE. 

The  mean  rate  and  the  several  prices  being  reduced  to 
the  same  denomination, — connect  with  a  contimied  line  each 
price  that  is  less  than  the  mean  rate  witK  one  or  more  that 
is  GREATER,  and  each  price  greater  than  the  mean  rate 
with  one  or  more  that  is  less. 

Write  the  di^'erence  between  the  mean  m/e,  or  price j  and 
the  price  of  each  simple  opposite  the  price  with  which  it  is 
connected;  (thus  the  difference  of  the  two  prices  in  each 
pair  will  be  mutually  exchanged;)  then  the  sum  of  the  differ^ 
ences,  standing  against  any  price^  will  express  the  relative 
QUANTITY  to  he  taken  of  that  price. 

By  attentively  considering  the  rule,  the  pupil  will  per- 
ceive, that  there  may  be  as  many  different  ways  of  mixing 
the  simples,  and  consequently  as  many  different  answers,  as 
there  are  different  ways  of  linking  the  several  prices. 

We  will  now  apply  the  rule  to  solve  the  last  question : — • 

OPERATIONS. 
Ihs. 
-2^  Or, 


—2)  (12 


2       ~2) 


/  &s. — - 

U2s 

Here  we  set  down  the  prices  of  the  simples,  one  directly 
linder  another,  in  order,  from  least  to  greatest,  as  this  is 
most  convenient,  and  write  the  mean  rate,  (10  s.)  at  the 
left  hand.  In  the  first  way  of  linking,  we  find,  that  we 
may  take  in  the  proportion  of  2  pounds  of  the  teas  at  8 
and  12  s.  to  1  pound  at  9  and  11  s.  In  the  second  way, 
we  find  for  the  answer,  3  pounds  at  8  and  11  s.  to  1  pound 
at  9  and  12  s. 

3.  What  proportions  of  sugar,  at  8  cents,  10  cents,  and 
14  cents  per  pound,  will  compose  a  mixture  worth  12  cent* 
per  pound  ? 

^715.  In  the  proportion  of  2  lbs.  at  8  and  10  cents  to  6 
lbs.  at  14  cents. 

Note,  As  these  quantities  only  express  the  proportions  of 
each  kind,  it  is  plain,  that  a  compound  of  the  same  mean 
price  will  be  formed  by  taking  3  times,  4  times,  one  half,  or 
any  proportion,  of  each  quantity.     Hence, 

When  the  quantity  of  one  simple  is  given,  after  findicg 


200  ALLIGATION.  IT  102. 

the  proportional  quantities,  by  the  above  rule,  we  may  say, 
As  the  VROPORTiofi XL  quantity  :  is  to  the  given  quantity:: 
so  is  each  of  the  other  proportional  quantities  :  to  the  re- 
quired quantities  .0/  each. 

4.  If  a  man  wishes  to  mix  1  gallon  of  brandy  worth 
16  s.  with  rum  at  9  s.  per  gallon,  so  that  the  mixture 
may  be  worth  11  s.  per  gallon,  how  much  rum  must  he 
use? 

Taking  the  differences  as  above,  we  find  the  proportions 
to  be  2  of  brandy  to  5  of  rum;  consequently,  1  gallon  of 
brandy  will  require  2^  gallons  of  rum.  Ans,  2^  gallons. 

5.  A  grocer  has  sugars  worth  7  cents,  9  cents,  and  12 
cents  per  pound,  which  ha  would  mix  so  as  to  form  a  com- 
pound worth  10  cents  per  pound ;  what  must  be  the  pro- 
portions of  each  kind  ? 

Am,  2  lbs.  of  the  first  and  second  to  4  lbs.  of  the  third  kind, 

6.  If  he  use  1  lb.  of  the  first  kind,  how  much  must  he  take 

of  the  others  ? if  4  lbs.,  what ? if  6  lbs.,  what  ^ 

if  10  lbs.,  what? if  20  lbs.,  what? 

Ans,  to  the  last^  20  lbs.  of  the  second,  and  40  of  the  third. 

7.  A  merchant  has  spices  at  16  d.  20  d.  and  32  d.  per 
pound ;  he  would  mix  5  pounds  of  the  first  sort  with  the 
others,  so  as  to  form  a  compound  worth  24  d.  per  pound ; 
how  much  of  each  sort  must  he  use  ? 

Arts,  5  lbs.  of  the  second,  and  7J  lbs.  of  the  third. 

8.  How  many  gallons  of  water,  of  no  value,  must  be 
mixed  with  60  gallons  of  rum,  worth  80  cents  per  gallon,  to 
reduce  its  value  to  70  cents  per  gallon  ?        Am,  S^  gallons. 

9.  A  man  would  mix  4  bushels  of  wheat,  at  $  1*50 
per  bushel,  rye  at  $146,  corn  at  $'75,  and  barley 
at  $  '50,  so  as  to  sell  the  mixture  at  $  '84  per  bushel ; 
how  much  of  each  may  he  use  ? 

10.  A  goldsmith  would  mix  gold  17  carats  fine  with 
some  19,  21,  and  24  carats  fine,  so  that  the  compound  may 
be  22  carats  fine ;  what  proportions  of  each  must  he  use  ? 

Ans,  2  of  the  3  first  sorts  to  9  of  the  last 

11.  If  he  use  1  oz.  of  the  first  kind,  how  much  must 
he  use  of  the  others  ?  What  would  be  the  quantity  of  the 
compound  ?  Am,  to  last^  7^  ounces. 

12.  If  he  would  have  the  whole  compound  consist  of  16 

oz.,   how  much  must  he  use  of  each  kind  ? if  of  30 

oz.,  how  much  of  each  kind  ? if  of  37^  oz.,  how  much  ? 

Ans,  to  the  last^  5  oz.  of  the  3  first,  and  22  j^  oz.  of  the  last 


If  102,  103.  DUODECIMALS.  201 

Hence,  when  the  quantity  of  the  compound  is  given,  wis 
may  say.  As  the  sum  of  the  proportional  quarttities,  found 
by  the  above  rule,  is  to  the  quantity  required,  so  is  each 
PROPORTIONAL  quantity^  found  by  the  rule,  to  the  required 
quantity  of  each. 

13.  A  man  would  mix  100  pounds  of  sugar,  some  at  8 
cents,  some  at  10  cents,  and  some  at  14  cents  per  pound,  so 
that  the  compound  may  be  worth  12  cents  per  pound;  how 
much  of  each  kind  must  he  use  ? 

We  find  the  proportions  to  be,  2,  2,  and  6.  Then,  2+2 
+  6  =  10,  and  C  2   :  20  lbs.  at    8  cts.  ) 

10   :    100    :  :  <  2   !  20  lbs.  at  10  cts.  V  Ans, 
(6    :  60  lbs.  at  14  cts.  ) 

14.  How  many  gallons  of  water,  of  no  value,  must  be 
mixed  with  brandy  at  $  1^20  per  gallon,  so  as  to  fill  a  ves- 
sel of  75  gallons,  which  may  be  worth  92  cents  per  gallon? 

Ans.  17^  gallons  of  water  to  57^  gallons  of  brandy. 

15.  A  grocer  has  currants  at  4  d.,  6  d.,  9d.  and  lid.  per 
lb. ;  and  he  would  make  a  mixture  of  240  bis.,  so  that  the 
mixture  may  be  sold  at  8  d.  per  lb. ;  how  many  pounds  of 
each  sort  may  he  take  ? 

Ans,  72,  24,  48,  and  96  lbs.,  or  48,  48,  72,  72,  &c. 
Note,  This  question  may  have  five  different  answers. 

QUESTIONS. 

1.    Wh?t  is    alligation?    2.  medial?    3.  — -  the, 

rule  for  operating  ?  4.  Wha.t  is  alligation  alternate  ?  5. 
When  the  price  of  the  mixture,  and  the  price  of  the  several 
simples,  are  given,  how  do  you  find  the  proportional  quanti- 
ties of  each  simple  ?  6.  When  the  quantity  of  one  simple  is 
given,  how  do  you  find  the  others  ?  7.  When  the  quantity 
of  the  whole  compound  is  given,  how  do  you  find  the  quan- 
tity of  each  simple  ? 


DUODliCIBSiL£S. 

IT  103.  Duodecimals  are  fractions  of  a  foot.  The  word 
is  derived  from  the  Latin  word  duodecimy  which  signifies 
twelve,  A  foot,  instead  of  being  divided  decimally  into  ten 
equal  parts,  is  divided  duodecimally  into  twelve  equal  parts, 


tali  MULTIPLICATION   OF   DUODECIMALS.         IT  103* 

called  inches,  or  primes^  marked  thus,  (').  Again,  each  of 
these  parts  is  conceived  to  be  divided  into  twelve  other  equal 
parts,  called  seconds^  (").  In  like  manner,  each  second  is 
conceived  to  be  divided  into  twelve  e»qual  parts,  called  thirds, 
('") ;  each  third  irto  twelve  equal  parts,  called  fourths^ 
{"")  ;  and  so  on  to  any  extent. 

In  this  way  of  dividing  a  foot,  it  is  obvious,  that 
1'  inchy  or  prime^  is  -----  •  xV  of  a  foot 
V^  second  is  -^  of  y'j,  -  -  -  iz:  y^  of  a  foot 
V  third  is -j^  of  1^  of  1^,  -  -  =z  yy^  of  a  foot 
V"  fourth  is  y^  of  -j^  of  -j^  of  y^,  z=  ^jj}-^  of  a  foot. 
r'^''  fifthisyVof  T^of  yVofiVof-iV,  =  ttVfs-IT  of  afoot,  &C- 

Duodecimals  are  added  and  subtracted  in  the  same  man- 
ner as  compound  numbers,  12  of  a  less  denomination  making 
1  of  a  greatery  as  in  the  following 

TABLB. 

12''"  fourths     make     V'  third, 
12'''   thirds    .     -     -     1''  second, 
12"    seconds      -     -     1'    inch  or  prime, 
12'     inches,  or  primes,  1     foot. 

^  Note.  The  marks,  ',  ",  '",  "",  &c.,  which  distinguish  the 
different  parts,  are  called  the  indices  of  the  parts  or  denomi- 
nations. 


MULTIPLICATION    OF    DUODECIMALS. 

Duodecimals  are  chiefly  used  in  measuring  surfaces  and 
solids. 

1.  How  many  square  feet  in  aboard  16  feet  7  inches  long, 
and  1  foot  3  inches  wide  ? 

Note.     Length  X  breadth  =  superficial  contents,  (IT  25.) 

OPERATION.  7  inches,  or  primes,  =  ^^  of  a 

Length    16     7'  ^^^*'  ^"^  ^  inches  =  y^  of  a  foot ; 

BrAdJh    \     ^'  consequently,  the  product  of  7'  X 

'  J_l__  3'  =  y%  of  a  foot,  that  is,  21" 

4     1'     9"  =  1'  and  9"  ;  wherefore,  we  set 

16     7'  down  the  9",  and  reserve  the  1' 

A       ^ — 1^1 '  ^  ^^  carried  forward  to  its  proper 

Ans.  20    8      9"  place.     To  multiply  16  feet  by  3' 


IT  103.         MITLTIPLICATION   OF    DUODECIMALS.  SOS 

is  to  take  -^  of  J/^  =  ff ,  that  is,  48' ;  and  the  1'  which  w« 
reserved  makes  49',  z=  4  feet  1' ;  we  therefore  set  down 
the  1',  and  carry  forward  the  4  feet  to  its  proper  place. 
Then,  multiplying  the  multiplicand  by  the  1  foot  in  the  mul- 
tiplier, and  adding  the  two  products  together,  we  obtain  the 
Answer,  20  feet,  8',  and  9". 

The  only  difficulty  that  can  arise  in  the  multiplication  of 
duodecimals  is,  in  finding  of  what  denomination  is  the  pro- 
duct of  any  two  denominations.  This  may  be  ascertained  as 
above,  and  in  all  cases  it  will  be  found  to  hold  true,  that  the 
product  of  any  two  denominations  will  always  be  of  the  denomi- 
nation denoted  by  the  sum  of  their  indices.  Thus,  in  the 
above  example,  the  sum  of  the  indices  of  7'  X  3'  is  "  ;  con- 
sequently, the  product  is  21" ;  and  thus  primes  multiplied 
by  primes  will  produce  seconds  ;  primes  multiplied  by  seconds 
produce  thirds  ;  fourths  multiplied  hy  fifths  produce  ninths,  &c. 

It  is  generally  most  convenient,  in  practice,  to  multiply  the 
multiplicand  first  by  the  feet  of  the  multiplier,  then  by  the 
inches,  &c.,  thus  :-^ 

A  16  ft.    X    1  ft.  =  16  ft.,   and  7'  X 

1  ft.  =  r.  Then,  16  ft.  X  3'  =  48' 
=  4  ft.,  and  7'  X  3'  =  21^'  =  1'  9". 
The  two  products,  added  together,  give 
for  the  Ajiswer,  20  ft.  8'  9',  as  before. 

20     8'     9" 

2.  How  many  solid  feet  in  a  block  15  ft.  8'  long,  1  ft.  C 
wide,  and  1  ft.  4'  thick  ? 

OPERATION. 
ft- 
Length,     15     8'  The  length  multiplied  by  the 

Breadth,     I     5'  breadth,  and  that  product  by  the 

thickness,   gives   the    solid    con' 
tents,  (IF  36.) 


16 

r 

1 

3' 

16 

7' 

4 

V 

9" 

15 

8' 

If 
te 

6 

6' 

4" 

22 

2' 

4" 

Thickness 

,    1 

4' 

22 

2i 

4" 

7 

4' 

9" 

4'" 

Am.     29     7'     1"     4" 


jfdi  MULTIPLICATION    OF    DUODECIMALS.        IT  104* 

From  these  examples  we  derive  the  following  Rule  : — 
Write  down  the  denominations  as  compound  numbers,  and 
in  multiplying  lemember,  that  the  product  of  any  two  de- 
nominations will  always  be  of  that  denomination  denoted  bj 
the  sum  of  their  indices, 

EXAMPLES  FOR  PRACTICE. 

5.  Hoyy  many  square  feet  in  a  &tpck  of  15  boards,  12  ft. 
8'  in  length,  and  13'  wide  ?  Ans,  205  ft.  10\ 

4.  What  is  the  product  of  371  ft.  2'  6"  multiplied  by 
181  ft,  V  9"?  Ans.  67242  ft.  10'  1''  4'"  6''". 

Note,  Painting,  plastering,  paving,  and  some  other  kinds 
of  work,  are  done  by  the  square  yard.  If  the  contents  in 
square  feet  be  divided  by  9,  the  quotient,  it  is  ^evident,  will 
be  square  yards. 

5.  A  man  painted  the  walls  of  a  room  8  ft.  2'  in  height, 
and  72  ft.  4'  in  compass ;  (that  io,  the  measure  of  all  its 
sides ;)  how  many  square  yards  did  he  paint  ? 

f  Ans.  65  yds.  5  ft.  8'  8''. 

*6.  There  is  a  room  plastered,  the  compass  of  which  i$ 
47  ft.  3',  and  the  height  7  ft.  6' ;  what  are  the  contents  ? 

Ans.  39  yds.  3  ft.  4'  6^'. 
\  7.  How  many  cord  feet  of  wood  in  a  load  8  feet  long,  4 
feet  wide,  and  3  feet  6  inches  high  ? 

Note,  It  will  be  recollected,  that  16  solid  feet  make  a 
cord  foot,  Ans.  7  cord  feet. 

8.  In  a  pile  of  wood  176  It.  in  length,  3  ft.  9'  wide,  and 
4  ft.  <y  high,  how  many  cords  ? 

Ans.  21  cords,  and  7-j^  cord  feet  over. 
*9.  How  many  feet  of  cord  wood  ii*  a  load  7  feet  long,  3 
feet  wide,  and  3  feet  4  inches  high  ?  and  what  will  it  come 
to  at  $  '40  per  cord  foot  ? 

Ans,  4 1  cord  feet,  and  it  w^ill  come  to  $  1'75. 
10.  How  much  wood  in  a  load  10  ft.  in  length,  3  ft,  9'  in 
width,  and 4 ft.  8'  in  height?  and  what  will  it  cost  at  $  1^92 
per  cord  ? 

Ans.  1  cord  and  2-f|  cord  feet,  and  it  will  come  to 
*2'62i.. 

^  104.  Remark.  By  some  surv^eyors  of  wood,  dimen- 
aions  are  taken  in  feet  and  decimals  of  a  foot.  For  this  purr 
pose,  make  a  rule  or  scale  4  feet  long,  and  divide  it  into  feet, 
and  each  foot  into  ten  equal  parts.     On  one  end  of  the  rule, 


IT  104,  105.  INVOLUTION-  205 

for  1  foot,  let  each  of  these  parts  be  divided  into  10  other 
equal  parts.  The  former  division  will  be  lOths,  and  the  lat- 
ter lOOtlis  of  a  foot.  Snch  a  rule  will  be  found  very  con- 
venient for  surveyors  of  wood  and  of  lumber,  for  painters, 
joiners,  &c. ;  for  the  dimensions  taken  by  it  beiug  in  feet  and 
decimals  of  a  fool,  the  casts  will  be  no  othe"  than  so  kiany 
operations  in  decimal  fractions. 

11.  How  many  square  feet  in  a  heartli  stone,  which,  by  a 
rule,  as  above  described,  measures  4'5  feet  in  lengtli,  and 
2^6  feet  >:i  width  ?  and  what  will  be  its  cost,  at  75  cents  per 
square  foot  ?  Ans,  ITT  feet;  and  it  will  cost  $8'775. 

f  12.  Hew  many  cords  in  a  load  of  wood  7'o  feet  in  length, 
3^6  feet  in  width,  and  4'8  feet  in  height  ?  Ans.  1  cord,ly%ft. 
^13.  How  many  cord  feet  in  a  load  cf  wood  10  feet  long, 
3^4  feet  wide,  and  3'5  feet  high  ?  Ans,  l-f^, 

QUESTIONS. 

1.  What  are  duodecimals?  2.  From  what  is  the  word 
derived?  3.  Into  how  many  parts  is  a  foot  usually  divided, 
and  what  tire  the  parts  called  ?  4.  What  are  the  other  de- 
nominations ?  5.  What  is  understood  by  the  indices  of  the 
denominations?  6.  In  what  are  duodecimals  chiefly  used? 
7.  How  are  the  contents  of  a  mrface  bounded  by  straight  lines 
found?  8.  How  are  the  contents  of  a  so/iVZ  found?  9.  How 
is  it  known  of  what  denomination  is  the  product  of  any  two 
denominations  ?  10.  How  may  a  scale  or  rule  be  formed 
for  taking  dimensions  in  feet  and  decimal  parts  of  a  foot  ? 


^  105.  Involution,  or  the  raising  of  powers,  is  the  mul- 
tiplying any  given  number  into  itself  continually  a  certain 
number  of  times.  The  products  tlnis  produced  arc  cafled 
the  powers  of  the  given  number.  The  number  itself  is  cUled 
the  first  power,  or  root.  If  the  first  power  be  multiplied  hy 
itselfy  the  product  is  called  the  second  power  or  square ;  if 
the  square  be  multiplied  by  the  first  power,  the  product  is 
called  the  third  power,  or  cuhe^  &c. ;  thus, 

5  is  the  root,  or  1st  power,  of  5. 
5X5=  25  is  the  2d  power,  or  square,  of  5,        =5^ 
5X5X5=125  is  the  3d  power,  or  cube,  of  5,  =5^ 

''>X5X5X5=:G25  is  the  Ithpower,  orbiquadrate,of 5,  =5^ 


506  INVOLUTION.  H  105. 

The  number  denoting  the  power  is  called  the  indeXj  or 
€Xp(r>icnt ;  thus,  5*  denotes  that  5  is  raised  or  involved  to 
the  4th  power. 

1.  What  is  the  square,  or  2d  power,  of  7  ?  Ans.  49. 

2.  What  is  the  square  of  30  ?  Ans,  900. 

3.  What  is  the  square  of  4000  ?  Ans,  16000000. 

4.  What  is  the  cube,  or  3d  power,  of  4  ?  Ans.  64. 

5.  What  is  the  cube  of  800  ?  Ans.  512000000. 

6.  Vv^hat  is  the  4th  power  of  60  ?  A.ns,  12960000. 

7.  What  is  the  square  of  1  ?      of  2  ?      of  3  ? 

of  4?  A71S.  1,  4,  9,  and  16. 

8.  What  is  the  cube  of  1  ?      of  2  ?      of  3  ? 

of  4  ?  Ans.  1,  8,  27,  and  64. 

9.  What  is  the  square  of  f  ?     of  |-  ?     of  |-  ? 

Ans.  I,  if,  and  ^. 

10.  What  is  the  cube  of  |?     of  |  ?      of  f  ? 

^ns. /y,f.\,andM. 

il.  Wliat  is  the  square  of  ^  ?     the  5th  povrer  of  i? 

A?is.  ^,  and  -gV. 

12.  W^hat  is  the  square  of  1'5  ?     the  cube  ? 

Ans.  2^25,  and  3'375. 

13.  What  is  the  6th  power  of  1^2  ?  Ans.  2'9859S4. 

14.  Involve  2^  to  the  4th  power. 

Note.  A  mixed  number,  like  the  above,  may  be  reduced 
to  an  improper  fraction  before  involving  :  thus,  2^  1=  |- ;  or 
it  may  be  reduced  to  a  decimal ;  thus,  2^  z=:  2'25. 

Ans. -^J>ry^  =1  2oiU. 

15.  What  is  the  square  of  4|-  ?  '  Ans.  J  |f  J^  =:  23|-|-. 

16.  What  is  the  value  of  7^,  that  is,  the  4th  power  of  7  ? 

Ans.  2401. 

17.  How  much  is  9^  ?     6^  ?     10^  ? 

Ans.  729,  7776,  10000 

18.  How  much  is  27  ?     3^  ? 4-'^?     5^  ? 

65  ?     103  ?  Ans.  to  last,  100000000. 

The  powers  of  the  nine  digits,  from  the  first  power  to  the 
iifth,  may  be  seen  in  the  following 

TAB1.E. 


Roots      -      or  1st  Powers  1  |  2  ]     3  |      4 

5 

6  1        7 1        8          9 

Squares        or  2d  Powers  1  j  4  |     9  |     IG 

25 

36  J       49  1       64         81 

Cubes     -     .or  3d  Powers  1  |  8  |  27  |     64 

125 

216  5     tm\     612      72^1 

Biquadrates  or  4th  Powers  1  |16     81     256 
Sursolids       or  5lh  Powers  1  |32  243  1024 

625! 
3125 

1296  1  ^401     4096     6661 
7776  116807  32768  50ai9 

If  107    EXTRACTION  OF  THE  SQUARE  ROOT.       207 


X:VO£UTZO]N< 

IT  106.  Evolution,  or  the  extracting  of  roots,  is  the  me- 
thod of  finding  the  root  of  any  power  or  number. 

The  rootj  as  we  have  seen,  is  that  number,  which,  by  a 
continual  multiplication  into  itself,  produces  the  given  power. 
The  square  root  is  a  number  which,  being  s<.|uared,  will  pro- 
duce the  given  number;  and  the  cube,  or  third  root,  is  a  num- 
ber which,  being  cubed  or  involved  to  the  3d  power,  will 
produce  the  given  nusiber:  thus,  the  square  root  of  144  is 
12,  because  12^  =i  144;  and  the  cxihe  root  of  343  is  7,  be- 
cause 7 2,  that  is,  7  X  7  X  7,  :ii=  343 ;  and  so  of  other  num- 
bers. 

Although  there  is  no  number  which  will  not  produce  a 
perfect  power  by  involatisn,  yet  there  are  many  numbers  of 
which  precise  roots  can  never  be  obtained.  But,  by  the 
help  of  decimals,  we  can  approximate,  or  approach,  towards 
the  root  to  any  assigned  degree  of  exactness.  Numbers, 
whose  precise  roots  cannot  be  obtained,  are  called  surd 
numbers,  and  those,  whose  roots  can  be  exactly  obtained,  are 
called  rational  numbers. 

The  square  root  is  indicated  by  this  character  .\/  placed 
before  the  number ;  the  other  roots  by  the  same  cliaracter, 
with  the  index  of  the  root  placed  over  it.  Thus,  the  square 
root  of  16  is  expressed  \/i6 ;  and  the  cube  root  of  27  is 
expressed   -v^27;  and  the  5th  root  of  7776,^^7776? 

When  the  power  is  expressed  by  several  numbers,  with 
the  sign  -f-  or  —  between  them,  a  line,  or  vinculum,  is  drawn 
from  the  top  of  the  sign  over  all  the  parts  of  it ;  thus,  the 
square  root  of  21  —  5  is  \/  21  —  5,  &c. 


EXTRiiCTZOnr    OF    TWSL    SQirARE 

HOOT. 

IT  107.  To  extract  the  square  root  of  any  number  is  to 
find  a  number,  which,  being  multiplied  into  itself,  shall  pro- 
duce the  given  number. 

1.  Suppesing  a  man  has  625  yards  of  carpeting,  a  yard 
wide,  what  is  the  length  of  one  side  of  a  squaie  room,  the 


208 


EXIRACTION  OF  THE  SQUARE  ROOT.    IF  lOt. 


iioor  of  v/hicli  the  carpetiiiof  will  cover  ?  tliat  is,  what  is  one 
side  of  a  square,  which  contains  625  square  yards  ? 

We  have  seen,  (11  85,)  that  the  contents  of  a  square  sur- 
face is  found  by  multiplying  the  leiigth  of  one  side  into  it- 
self, that  is,  by  raising  it  to  the  second  power;  and  hence, 
having  the  contents  (625)  given,  we  must  extract  its  square 
root  to  find  one  side  of  the  room. 

This  we  must  do  by  a  sort  of  trial  :    and, 

1st.  We  will  endeavour  to  ascertain  how  many  figures 
t':ere  will  be  in  the  root.  This  we  can  easily  do,  by  point- 
ing off  the  number,  from  units,  into  periods  of  two  figures 
each ;  for  the  square  of  any  root  always  contains  just  fwice  as 
many,  or  one  figure  less  than  twice  as  many  figures,  as  are 
in  the  root ;  of  which  truth  the  pupil  may  easily  satisfy  him- 
self by  trial.     Pointiug  off  the  number,  we  find,  that  the 

root  will  consist  oHwo  figures, 


OPERATION. 

C25(2 

4 

225 


Fig.  I. 


a  ten  i^nd  a  unit. 

2d.  We  will  now  seek  for 
the  first  figure,  that  is,  for 
the  tens  of  the  root,  and  it  is 
plain,  that  we  must  extract  it 
from  the  left  hand  period  6, 
(hundreds.)  The  greatest 
square  in  6  (hundreds)  we 
find,  by  trial,  to  be  4,  (hun- 
dreds,) the  root  of  which  is  2, 
(tens,  1=  20;)  therefore,  we 
set  2  (tens)  in  the  root.  The 
root,  it  will  be  recollected,  is 
07ie  side  of  a  square.  Let  us, 
then,  form  a  square,  (A,  Fig. 
1.)  each  side  of  which  shall  be 
supposed  2  ten  -,  =  20  yards, 
expressed  by  the  root  now 
obtained. 

The  contents  of  this  square  are  20  X  20  z=  400  yards,  now 
disposed  of,  and  which,  consequentl};,  are  to  be  deducted  from 
the  whole  number  of  yards,  (625,)  leaving  225  yaids.  This 
dedr.ction  is  most  readily  performed  by  subtracting  the  square 
number  4,  (hundreds,)  or  the  square  of  2,  (the  figure  in  the 
root  already  found,)  from  the  period  6,  (hundreds,)  and  bring- 
ing down  the  next  period  by  the  side  of  the  remainder, 
making  225,  as  before. 


IT  107.         JEXTRACTION    OF    THE    gqUAKE    ROOT. 


209 


3d.  The  square  A  is  now  to  be  enlarged  by  the  addition 
of  the  225  remaining  yards ;  and,  in  order  that  llie  figure 
may  retain  its  square  form^  it  is  evident,  the  addition  must 
be  made  on  tivo  sides.  Now,  if  the  225  yards  be  divided  by 
the  length  of  the  two  sides,  (20  -f-  20  =  40,)  the  quotient 
will  be  the  breadth  of  this  new  addition  of  225  yards  to  the 
sides  c  d  and  b  c  of  the  square  A. 

But  our  root  already  found,  ^i^^tens,  is  the  length  of  one 
side  of  the  figure  A ;  we  thergfor.e  take  double  this  root,  iir  4 
tens,  for  a  divisor. 


OPERATION— CONTINUED. 

625(25  "^ 

4 

45)225 
225 


Fig. 

SOyds. 


II. 


5  yds. 


B 

20 
5 

100 

5 

D  J 

25 

d 

A 

20 
20 

400 

C 

c 

20 
5 

100 

a 

b 

20  yds. 


5  yds. 


,  The  divisor,  4,  (tens,) 
is  in  reality  40,  and  we 
are  to  seek  how  many 
times  40  is  contained  in 

.  225,  or,  which  is  the 
same  •  thing,  we  may 
seek  how  many  times 
4  (tens)  is  contained  in 

*22,  (tens,)  rejecting  the 
right  hand  figure  of  the 
dividend,  because  we 
have  rejected  the  cipher 
in  the  divisor.  We  find 
our  quotient,  that  isj  the 
breadth  of  the  addition, 
to  be  5  yards ;  but,  if 
we  look  at  Fig.  II.,  we 
shall  perceive  that  this 
addition  of  5  yards  to  the 
two  sides  does  not  com- 
plete the  square ;  for 
there  is  still  wanting,  in 
the  corner  D,  a  small 


square,  eac»i  side  of 
which  is  equal  to  this  last  quotient,  5  ;  we  must,  therefore, 
add  this  quotient,  5,  to  the  divisor,  40,  that  is,  place  it  at  the 
right  hand  of  the  4,  (tens,)  making  it  45;  and  then  the  whole 
divisor,  45,  multiplied  by  the  quotient,  5,  wiil  give  the  con- 
tents of  the  whole  addition  around  the  sides  of  the  figure  A, 
which,  in  this  case,  being  225  yards,  the  same  as  our  divi- 
dend, we  have  no  remainder,  and  the  work  is  done.  Con- 
•equently,  Fig.  II.  represents  the  floor  of  a  square  room,  25 


210  EXTRACTION    OF    THE    SC^UARE   ROOT.        iT  lOt. 

yards  on  a  side,  which  625  square  yards  of  carpeting  will 
exactly  cover. 

The  proof  may  be  seen  by  adding  together  the  several 

parts  cf  the  figure,  thus  : — 

The  square  A  contains  400  yards. 

The  figure  B 100  Or  we  may  prove  it 

C 100 by  involution,    thus  : — 

J^...  1> 25 25  X  25  =  625,  as  be- 

Proo/,~625 ^''''^* 

From  this  example  and  illustration  we  derive  the  follomng 
general 

RULE 

FOR  THE  EXTRACTION  OF  THE  SQUARE  ROOT. 

I.  Point  off  the  given  number  into  periods  of  two  figures 
each,  by  putting  a  dot  over  the  units,  another  over  the  hun- 
dreds, and  so  on.  These  dots  show  the  number  of  figures 
of  which  the  root  will  consist. 

II.  Find  the  greatest  square  number  in  the  left  hand  pe- 
riod, and  write  its  root  as  a  quotient  in  division.  Subtract 
the  square  number  from  the  left  hand  period,  and  to  the  re- 
mainder bring  down  the  next  period  for  a  dividend. 

III.  Double  the  root  already  found  for  a  divisor ;  seek  how 
many  times  the  divisor  is  contained  in  the  dividend,  except- 
ing the  right  hand  figure,  and  place  the  result  in  the  root, 
and  also  at  the  right  hand  of  the  divisor;  multiply  the  di- 
visor, thus  augmented,  by  the  last  figure  of  the  root,  and 
subtract  the  product  from  the  dividend  ;  to  the  remainder 
bring  down  the  next  period  for  a  new  dividend. 

IV.  Double  the  root  already  found  for  a  new  divisor,  and 
continue  the  operation  as  before,  until  all  the  periods  are 
brought  down. 

Note  1.  If  v/e  double  the  right  hand  figure  of  the  last 
divisor,  we  shall  have  the  double  of  the  root. 

Note  2.  As  the  value  of  figures,  whether  integers  or 
decimals,  is  determined  by  their  distance  from  the  place 
of  units,  so  we  must  always  begin  at  unit's  place  to  point  off 
the  given  number,  and,  if  it  be  a  mixed  number,  we  must 
point  it  off  both  ways  from  units,  and  if  there  be  a  deficiency 
in  any  period  of  decimals,  it  may  be  supplied  by  a  cipher. 
li  is  plain,  the  rcfn  n  urt  ^Iv   ;  -  'consist  of  so  many  integer* 


ir  108.    EXTRACTION  OF  THE  SQUARE  ROOT.      211 

and  decimals  as  there  are  periods  beloDging  to  each  in  the 
given  number. 

exa3ipl.es  for  practice. 

2.  What  is  the  square  root  of  1034265G  ? 
OPERATION. 

10342656  (  3216,  Am, 
9 

62  )  134 
124 


641  )  1026 
641 


6426  )  38556 
38556 


3.  What  is  tlie  square  root  of  43264  ? 
OPERATION. 


43264  (  208,  Ans. 
4 


408  )  3264 
3264 


4.  What  is  the  square  root  of  998001  ?  Atuf.  999 

5.  What  is  the  square  root  of  234'09.?  Ans,   15^3. 

6.  What  is  tlie  square  root  of  964*5192360241  ? 

Am.  31*05671. 

7.  What  is  the  square  root  of  '001296  >  Ans,  '036. 

8.  What  is  tlie  square  root  of  '2916  ?  Ans.  '54. 

9.  What  is  the  square  root  of  36372961  ?  Am.  6031. 

10.  What  is  the  square  root  of  164  ?  Ans,  12'8  +. 

ir  103.  In  this  last  example,  as  there  was  a  remainder, 
after  bringing  down  all  the  figures,  we  continued  the  opera- 
tion to  decimals,  by  annexing  two  ciphers  for  a  new  period, 
and  thus  we  may  continue  the  operation  to  any  assigned  de- 
gree of  exactness ;  but  the  pupil  w  ill  readily  perceive,  that 
he  can  never,  in  this  manner,  obtain  the  precise  root ;  for  the 
last  figure  in  each  dividend  will  always  be  a  cipher,  and  the 


212  SUPPLEMENT    TO    THE    SQUARE    BOOT.         IT   108. 

last  figure  in  eacli  divisor  is  the  same  as  the  last  quotient 
figure  ;  but  no  one  of  the  nine  digits,  multiplied  into  itself^ 
produces  a  number  ending  with  a  cipher ;  therefore,  what- 
ever be  the  quotient  figure,  there  will  still  be  a  remainder. 

11.  What  is  the  square  root  of  3  ?  Ans,   1*73  ■ 

12.  What  is  the  square  root  of  10  ?  Ans,  346  ■ 

13.  What  is  the  square  root  of  184'2  ?  Ans.   13'57+. 

14.  What  is  the  square  root  off? 

Note.  We  have  seen,  (U  105,  ex.  9,)  that  fractions  are 
squared  by  squaring  hath  the  numerator  and  the  denomina- 
tor. Hence  it  follows,  that  the  square  root  of  a  fraction  is 
found  by  extracting  the  root  of  the  numerator  and  of  the  de- 
nominator.    The  root  of  4  is  2,  and  the  root  of  9  is  3. 

Ans.  J. 

15.  "\Yhat  is  the  square  root  of /^?  Ans.  ^ 

16.  What  is  the  square  root  of  y\j%^?  Ans.  -^ 

17.  What  is  the  square  root  of  -f^^?  Ans.  ^^  =  f, 

18.  What  is  the  square  root  of  20 J  ?  Ans.  4^. 
When   the   numerator   and   denominator   are    not    exact 

squares^  the  fraction  may  be  reduced  to  a  decimal,  and  the 
approximate  root  found,  as  directed  above. 

19.  What  is  the  square  root  of  f  zn  '75  ?       Ans.  '866  +. 

20.  What  is  the  square  root  of  f  f  ?  Ans.  '912  -|-. 


SUPPI^SMEH^T  TO  THE  SQUARE   ROOT. 
QUESTIOIVS. 

1.  W^hat  is   involution  ?     2.  What   is   understood   by  a 

power  ?    3.  the  first,  the  second,  the  third,  the  fourth 

power  ?  4.  W^hat  is  the  index,  or  exponent  ?  5.  How  do 
you  involve  a  number  to  any  required  power  ?  6.  What  is 
evolution  ?  7.  What  is  a  root?  8.  Can  the  precise  root  of  all 

numbers  be  found  ?     9.  What  is  a  surd  number?  10. a 

rational?  11.  What  is  it  to  extract  the  square  root  of  any 
number?  12.  Why  is  the  given  sum  pointed  into  periods  of 
two  figures  each  ?  13.  Why  do  we  double  the  root  for  a 
divisor  ?  14.  Why  do  we,  in  dividing,  reject  the  right  hand 
figure  of  the  dividend  ?  15.  Why  do  we  place  the  quotient 
figure  to  the  right  hand  of  the  divisor  ?     16.  How  may  we 


fr   108.         SUPPLEMENT    TO   THE    SQUARE    ROOT.  213 

prove  the  work?  17.  Why  do  we  point  off  mixed  numbers 
both  ways  from  units?  18.  When  there  is  a  remainder, 
how  may  we  conthiue  the  op.eratiou  ?  19.  Why  can  we 
never  obtain  the  precise  root  of  surd  numbers  ?  20.  How 
do  we  extract  the  square  root  of  vulgar  fractions  ? 


EXERCISES. 

1.  A  general  has  4096  men  ;  how  many  must  he  place  in 
rank  and  i>le  to  form  them  into  a  square  ?  Ans,  64. 

2.  If  a  square  field  contains  2025  square  rods,  how  m?.ny 
rods  does  It  measure  on  each  side  ?  A7is.  45  rods. 

3.  How  many  trees  in  each  row  of  a  square  orchard  con- 
taining 5625  trees  ?  Ans.  75. 

4.  There  is  a  circle,  whose  area,  or  superficial  contents, 
is  5134  feet ;  what  will  be  the  length  of  the  side  of  a  square 
of  equal  area  ?  \/5184  nz  72  feet,  Ans. 

5.  A  has  two  fields,  one  containing  40  acres,  and  the  other 
containing  50  acres,  for  which  B  offers  him  a  square  field 
containing  the  same  number  cf  acres  as  both  of  these ;  how 
many  rods  must  each  side  of  this  field  measure  ? 

Ans.  120  rods. 

6.  If  a  certain  square  field  measure  20  rods  on  each  side, 
how  much  will  the  sidii  of  a  square  field  measure,  contain- 
ing 4  times  as  much  ?  V'20  X  20  X  4  zn  40  rods,  Ans. 

7.  If  the  side  of  a  square  be  5  feet^  what  will  be  the  side 

of  one  4  times  as  large  ? 9  times  as  large  ?  16 

times  as  large  ?  25  times  as  large  ?  — —  36  times  as 

large  ?  Answers,  10  ft. ;   15  ft. ;  20  ft. ;  25  ft. ;  and  30  ft. 

8.  It  is  required  to  lay  out  288  rods  of  land  in  the  form  of 
a  parallelogram,  which  shall  be  twice  as  many  rods  in  length 
as  it  is  in  width. 

Note.  If  the  field  be  divided  in  the  middle,  it  will  form 
two  equal  squares. 

Ans.  24  rods  long,  and  12  rods  wide. 

9.  I  would  set  out.  at  equal  distances,  784  apple  trees,  so 
that  my  orchard  may  be  4  times  as  long  as  it  is  broad  ;  how 
many  rows  of  trees  must  I  have,  and  how  many  trees  in 
each  row  ?  Ans.  14  rows,  and  56  trees  in  each  row. 

10.  There  is  an  oblong  piece  of  land,  containing  192  square 
rods,  of  which  the  width  is  |-  as  much  as  the  lengtli ;  re- 
quired its  dimensions.  Ans.  IC  by  12. 


SI 4  SUl»PLJBJ^Nt'    TO    THE    SQUARE    ROOT.         If   109» 

\ 

11.  There  is  a  circle,  whose  diameter  is  4  inches ;  what  is 
the  diameter  of  a  circle  9  times  as  large  ? 

Note,  The  areas  or  contents  of  circles  are  in  proportion 
to  the  squares  of  their  diameters^  or  of  their  circumferences. 
Therefore,  to  find  the  diameter  required^  square  the  given 
diameter,  multiply  the  square  by  the  given  ratio,  and  the 
square  root  of  the  product  will  be  the  diameter  required. 

/\/4  X  4  X  9  =  12  inches,  Ans. 

12.  There  are  two  circular  ponds  in  a  gentleman's  pleasure 
ground ;  the  diameter  of  the  less  is  100  feet,  and  the  greater 
is  3  times  as  large  ;  what  is  its  diameter  ?     Ans,  173'2-|-  feet 

13.  If  the  diameter  of  a  circle  be  12  inches,  what  is  the 
diameter  of  one  -^  as  large  ?  Ans,  6  inches. 

IT  109.  14.  A  carpenter  has  a  large  wooden  square  ;  one 
part  of  it  is  4  feet  long,  and  ^he  other  part  3  feet  long  ;  what 
is  the  length  of  a  pole,  which  will  just  reach  from  one  end  to 
the  otlier  ? 

A  Note,     A    figure    of  3 

sides  is  cilled  a  triangle, 
^  and,  if  one  of  the  corners 

"  be  a  square  corner^  or  rigkt 

angle^  like  the  angle  at  B 
in  the  annexed  figure,  it  is 
called  a  right-angled  trian- 
gle^ of  which  the  square 
of  the  longest  side,  A  C, 
(called  the  hypotenuse,) 
is  equal  to  the  sum  of  the  squares  of  the  other  two  sides,  A  B 

and  B  C.  

42  —  16,  and  3^  —  9  ;  then,  \/9  +  16  1=  5  feet,  Ans, 

15.  If,  from  the  corner  of  a  square  room,  6  feet  be  mea- 
sured off  one  way,  and  8  feet  the  oiher  way,  along  the  sides 
of  the  room,  what  will  be  the  length  of  a  pole  reaching  from 
point  to  point  ?  Ans,  10  feet. 

16.  A  wall  is  32  feet  high,  and  a  ditch  before  it  is  24  feet 
wide ;  what  is  the  length  of  a  ladder  that  will  reach  frcm  the 
top  of  the  wall  to  the  opposite  side  of  the  ditch  ? 

Ans,  40  feet 

17.  If  the  ladder  be  40  feet,  and  the  wall  32  feet,  what  is 
tlie  width  of  ih^  ditch  ?  Ans,  24  feet 

18.  The  ladder  and  dkch  given,  required  the  wall. 

Am,  32  feet 


/ 

.^^ 

X--" 

/ 

/ 

/ 

c 

Baso. 

B 

9 


T  110.     EXTRACTION  OF  THE  CUBE  ROOT.        215 

^19.  The  distance  between  the  lower  ends  of  two  equal 
rafters  is  32  feet,  and  the  height  of  the  ridge,  above  the  beam 
on  which  they  stand,  is  12  feet;  required  the  length  of  each 
rafter.  Ajis.  20  feet. 

20.  There  is  a  building  30  feet  in  length  and  22  feet  in 
width,  and  the  eaves  project  beyond  the  wall  1  foot  on  every 
side  ;  the  roof  terminates  in  a  point  at  the  centre  of  the 
building,  and  is  there  supported  by  a  post,  the  top  of  which 
is  10  feet  above  the  beams  on  which  the  rafters  rest;  what 
is  the  distance  from  the  foot  of  the  post  to  the  corners  of  the 
eaves  ?  and  what  is  the  length  of  a  rafter  reaching  to  the 

middle  of  one  side  1  a  rafter  reaching  to  the  middle  of 

one  end  ?  and  a  rafter  reaching  to  the  comers  of  the  eaves  ? 

An^oers,  in  order,  20  ft. ;   15'62  +  iX, ;   18^86  -f  ft. ;  and 
22^36  +  ft. 
•)    21.  There  is  a  field  800  rods  long  and  600  rods  wide  ; 
^  what  is  the  distance  between  two  opposite  corners  ? 

Ahs,  1000  rods. 

22.  There  is  a  square  field  containing  90  acres  ;  how 
many  rods  in  length  is  each  side  of  the  field  ?  and  how  many 
rods  apart  are  the  opposite  corners  ? 

Answers,  120  rods  ;    and  169'7 -f- rods. 

23.  There  is  a  square  field  containing  10  acres;  what  dis- 
tance is  the  centre  from  each  co^^ier  ?      Am,  28'28  +  rods. 


&ZTRJLCT£0»T   OF    THIS   CUBE 

ROOT. 

IT  110.  A  solid  body,  having  six  equal  sides,  and  each  of 
the  sides  an  exact  square,  is  a  cube,  and  the  measure  in 
length  of  one  of  its  sides  is  the  root  of  that  cube ;  for  the 
length,  brcfidth  and  thickness  of  such  a  body  are  all  alike  ;  con- 
sequently, the  length  of  one  side,  raised  to  the  3d  power, 
^ves  the  solid  contents.     (See  IT  36.) 

Hence  it  follows,  that  extracting  tne  cube  root  of  any  num- 
ber of  feet  is  finding  the  length  of  one  side  of  a  cubic  bo- 
dy, of  which  the  whole  contents  will  be  equal  to  the  given 
number  of  feet.  ^ 

1.  What  are  the  solid  contents  of  a  cubic  block,  of  which 
each  side  measures  2  feet  ?     Ans.  23-—  2X2X2r=8  feet 

2.  How  many  solid  feet  in  a  cubic  block,  n^easuring  6  feejt 
Oft  each  side?  Aiv3,  6'^'  z=z  125  feet. 


^16 


EXTRACTION  OF  THE  CUBE  ROOT. 


ffllO. 


3.  How  many  feet  in  length  is  eacli  side  of  a  cubic  block, 
containing  125 solid  feet?  Ans.  /v/125  =  5  feet. 

Note.  The  root  may  be  found  by  trial. 

4.  What  is  the  side  of  a  cubic  block,  containing  64  solid 

feet? 27  solid  feet?-- — 216  solid  feet? 512  solid 

feet  ?  A7iswers,  4  ft. ;  3  ft. ;  6  ft. ;  and  8  ft 

5.  Supposing  a  man  has  13824  feet  of  timber,  in  separate 
blocks  of  1  cubic  foot  each ;  he  wishes  to  pile  them  up  in 
a  cubic  pile ;  what  will  be  the  length  of  each  side  of  such 
a  pile  ? 

It  is  evident,  the  answer  is  found  by  extracting  the  cube 
root  of  13S24  ;  but  this  number  is  so  huge,  that  we  cannot 
so  easily  fmd  the  root  by  trial  as  in  the  former  examples  ; — 
We  will  endeavour,  however,  to  do  it  by  a  sort  of  trial ;  and, 
1st.  We  will  try  to  ascertain  the  number  of  figures,  of 
which  the  root  will  consist.  This  we  may  do  by  pointing 
the  number  off  into  periods  o(  three  figures  each  (IT  107,  ex.  1.) 

Pointing  off,  we  see,  the 
root  will  consist  of  two  figures, 
a  ten  and  a  tmt.  Let  us,  then, 
seek  for  the  first  figure,  or 
tens  of  the  root,  which  must 
be  extracted  from  the  left 
hand  period,  13,  (thousands.) 
The  greatest  cube  in  13 
(thousands)  we  find  by  trial, 
or  by  the  table  of  powers,  to  be 
8,  (thousands,)  the  root  of 
which  is  2,  (tens;)  therefore, 
we  place  2  (tens)  in  the  root. 
The  root,  it  will  be  recollect- 
ed, is  one  side  of  a  cube.  Let 
us,  then,  form  a  cube,  (Fig.  1.) 
each  side  of  which  shall  be 
supposed  20  feet,  expressed 
by   the    root    now    obtained. 

The  contents  of  this  cube  are 

Sim  feet,  Contents.  20  X  20  X  20  ~  8000  solid  feet, 

which  are  now  disposed  of,  and  which,  consequently,  are  to 
be  deducted  from  the  whole  number  of  feet,  13824.  8000 
taken  from  13824  leave  5824  feet.  This  deduction  is  most 
readily  performed  by  subtracting  the  cubic  number,  8,  or 
the  cube  of  2,  (the  figure  of  the  root  already  fouud,)  from 


OPERATION. 

13824(2 
8 

■~5824 


Fig.  L 


IT  110. 


EXTRACTION   or      HB   CUBE   ROOT. 


217 


the  period  13,  (thousands,)  and  bringing  down  the  next  pe- 
riod by  tlw  side  of  the  remainder,  making  5824,  as  before. 
2d.  The  cubic  pile  A  D  is  now  to  be  enlarged  by  the  ad- 
dition of  5824  solid  feet,  and,  in  order  to  preserve  the  cu'bic 
form  of  the  pile,  the  addition  must  be  made  on  one  half  of 
its  sides,  that  is,  on  3  sides,  a,  6,  and  c.  Now,  if  the  5824 
solid  feet  be  divided  by  the  square  contents  of  these  3  equal 
sides,  that  is,  by  3  times,  (20  X  20  —  400)  =  1200,  the  quo- 
tient will  be  the  thickness  of  the  addition  made  to  each  of 
tlie  sides  a,  ^,  c.  But  the  root,  2,  (tens,)  already  found,  is 
tlie  length  of  one  of  these  sides ;  we  therefore  square  the 
root,  2,  (tens,)  zz:  20  X  20  i=  400,  for  the  square  contents  of  one 
side,  and  multiply  the  product  by  3,  the  number  of  sides, 
400  X  3  =  1200 ;  or,  which  is  the  same  in  effect,  and  more 
convenient  in  practice,  v/e  may  square  the  2,  (tens  J  and  mul- 
tiply the  product  by  300,  thus,  2  X  2  =z  4,  and  4  X  300  ==  1 200, 
for  the  divisor,  as  before. 

The  divisor,  1200,  is  con- 
tained in  the  dividend  4  times ; 
consequently,  4  feet  is  the 
thickness  of  the  addition  made 

to  each  of  the  three  sides,  a, 

Dim^or,  1200)5824  Dividend,  Z>,  c,  and  4  X  1200  nz  4800,  is 


OPERATION— CONTINUED. 

13824(24  Root. 
8 


4800 

960 

64 

5824 
0000 


Fig.  II. 


the  solid  feet  contained  m 
these  additions;  but,  if  we 
look  at  Fig.  II.,  w^e  shall  per- 
ceive, that  this  addition  to  the 
3  sides  does  not  complete  the 
cube ;  for  there  are  deficiencies 
in  the  3  corners  w,  tz,  n.  Now 
the  length  of  each  of  these 
deficiencies  is  the  same  as  the 
length  of  each  side^  that  is,  2 
(tens)  =:  20,  and  their  ?vidth 
and  thickness  are  each  equal  to 
the  last  quotient  figure,  (4) ; 
their  contents,  therefore,  or 
the  number  of  feet  required  to 
fill  these  deficiencies,  will  be 
found  by  multiplyin^the  square 
of  the  last  quotient  figure,  (4^) 
nz  16,  by  the  length  of  all  the 
deficiencies,  that  is,  by  3  timei 
T 


218 


EXTRACnOir   OF   THE   CUBE    ROOT* 


IT  110. 


the  length  ©f  each  side,  which  is  expressed  by  the  former 
quotient  figure,  2,  (tens.)  3  times  2  (tens)  are  6  (tens)  = 
60 ;  or,  what  is  the  same  in  effect,  and  more  convenient  in 
practice,  we  may  multiply  the  quotient  figure,  2,  (tens,)  by 
30,  thus,  2  X  30  =  60,  as  before  ;  then,  60  X  16  =  960,  con- 
tents of  the  thri.e  deficiencies  «,  ??,  ii. 

Looking  at  Fig.  III.,  we 
perceive  there  is  stiil  a  de- 
ficiency in  the  corner  where 
the  last  blocks  meet.  This 
deficiency  is  a  cube,  each 
side  of  which  is  equal  to  the 
last  quotient  figure,  4.  The 
cube  of  4,  therefore,  (4  X  4 
X  4  —  64,)  will  be  the  solid 
contents  oi'this  corner,  which 
in  Fig.  IV.  is  seen  filled. 

Now,  the  sum  of  thesft  sev- 
eral additiolis,  viz.  4800  + 
960  +  64  —  5824,  will  make 
the  subtrahend,  wbich,  sub- 
tracted from  the  dividend, 
leaves  no  remainder,  and  the 
work  is  done. 

Fig.  IV.  shows  the  pile 
which  13824  solid  blocks  of 
one  foot  each  would  make, 
when  laid  together,  and  the 
root,  24,  sbows  the  length  of 
one  side  of  the  pile.  The 
correctness  of  the  work  may 
be  ascertained  by  cubing  the 
side  now  found,  24 »,  thus,  24 
X    24  X   24  :z=   13S24,  the 


2i  feet. 


given  number;  or  it  may   be   proved   by  adding  together 
^le  contents  of  all  the  several  parts,  thus. 

Feet 

8000  =:  contents  of  Fig.  I. 
4800  =  addition  to  the  sides  a,  Z>,  and  c,  Fig.  I. 
960  z=  addition  to  fill  the  deficiencies  w,  ?i,  w.  Fig.  II. 
^  =  addition  to  fill  the  corner  €,  e,  c,  Fig.  IV. 

13824  =  contents  of  the  whole  pile,  Fig.  IV.,  24  feet  on 
«ch,sida 


i^  1 10.     KKTBACriON  OF  THE  CUBE  ROOT.       2TS 

From  the  foregoing  example  and  Ulustraiion  we  derive  ike 
following 

FOR  EXTHACTING  THE  CUBE  ROOT. 

I.  Separate  the  given  number  into  periods  of  three  figures 
each,  by  putting  a  point  over  the  unit  figure,  and  every  third 
fig«re  beyond  the  place  of  units. 

II.  Find  the  greatest  cube  in  the  left  hand  period,  and  pu* 
its  root  in  the  quotient. 

III.  Subtract  the  Cube  thus  found  from  the  said  period, 
and  to  the  remainder  bring  down  the  next  period,  and  call 
this  ihe  dimdeiid, 

IV.  Multiply  the  square  of  the  quotient  by  300,  calling  in 
the  divisor. 

V.  Seek  how  many  times  the  divisor  may  be  had  in  the 
dividend,  and  place  the  result  in  the  root;  then  multiply 
the  divisor  by  this  quotient  figure,  and  write  the  product 
under  the  dividend. 

VI.  Multiply  the  square  of  this  quotient  figure  by  the 
former  figure  or  figures  of  the  root,  and  this  product  by  30, 
and  place  the  product  under  the  last ;  under  ail  write  the 
cube  of  this  quotient  figure,  and  call  their  amount  the  sub- 
trahend, ■; 

VII.  Subtract  the  subtrahend  from  the  dividend,  and  to  tha 
remainder  bring  down  the  next  period  for  a  new  dividend, 
with  which  proceed  as  before ;  and  so  on,  till  the  whole  is 
finished. 

Note  1.  if  it  happens  that  the  divisor  is  not  contained  in 
the  dividend,  a  cipher  must  be  put  in  the  root,  and  the 
next  period  brought  down  for  a  dividend. 

Note  2.  The  same  rule  must  be  observed  for  continuing 
the  operation,  and  pointing  oiF  for  decimals,  as  in  the  square 
root. 

Note  3.  The  pupil  will  perceive  that  the  number  which 
we  call  the  dioisor,  whea  multiplied  by  the  last  quotient 
figure,  does  not  produce  so  large  a  number  as  the  real  sub- 
tnJiend ;  hence,  the  figure  in  the  root  must  frequently  be 
Mttaller  than  the  quotient  figure. 


£80        SUPPLEMENT  TO  THE  CUBE  ROOT.     IT  110 

exampl.es  for  practice. 

^  6.  WTiat  is  the  cube  root  of  1860867? 
OPERATION. 

1860867(123  Ans. 

1 


1 2  X  300  =  300  )  860  first  Dividend, 

600 

22  X  1  X  30  HZ     120 

23      1=         8 

728  first  Subtrahend. 

122  X  300  =  43200  )  132867  second  Dividend 

129600 
32  X  12  X  30  =  3240 
^  33  =  27 


132867  second  Svbiraheiid, 

000000 

^7.  What  is  the  cube  root  of  373248  ?  Ans.  72. 

8.  What  is  the  cube  root  of  21024576  ?  Ans,  276. 

9.  What  is  the  cube  root  of  84^604519  ?  Am.  4*39. 

10.  What  is  the  cube  root  of  ^000343  ?  A)is.  '07. 

11.  What  is  the  cube  root  of  2  ?  Am.  1'25  -}-. 

12.  What  is  the  cube  root  of  ^^  ■:  Am.  §. 
Note.  See  IT  105,  ex.  10,  aud  IT  108,  ex.  14. 

13.  What  is  the  cube  root  of  ^|  ?  Am.  | 

14.  What  is  the  cube  root  of  ^V^g-  ?  ^R5.  iV 

15.  Wliat  is  the  cube  root  of  -^^-^  ?  Am.  425  +• 

16.  What  is  the  cube  root  of  y^^-  ?  Am.  \. 


SXyPPliEMSNT    TO   THE    CUBE   HOOT. 

QUESTIONS. 

1.  What  is  a  cube  }  2.  What  is  understood  by  the 
ciibe  root  ?  3.  What  is  it  to  extract  the  cube  root  ? 
4.  Why  is  the  square  of  the  quotient  multiplied  by  300 
for  a  divisor  ?  5.  Why,  in  finding  the  subtrahend,  do 
we  multiply  the  square  of  the  last  quctient  figure  by  30 
times  the  former  figure  of  the  root.^  6.  Why  do  we 
cube  the  quotient  figure  ?  7.  How  do  we  prove  the 
operation } 


H  111c  SUPPLEMENT   TO   THE   CtJBE   ROOT.  2Z\ 


EXERCISES. 

i.  What  is  the  side  of  a  cubical  mound,  equal  to  one  288 
feet  long,  216  feet  broad,  and  48  feet  high  ?  Am,  144  feet. 
"^  2.  There  is  a  cubic  box,  one  side  of  which  is  2  feet ;  how 
many  solid  feet  does  it  contain  ?  Ans.  8  feet. 

3.  How  many  cubic  feet  in  one  8  times  as  large  ?  and 
what  would  be  the  length  of  one  side  ? 

Alls,  64  solid  feet,  and  one  side  is  4  feet. 

4.  There  is  a  cubical  box,  one  side  of  which  is  5  feet ; 
what  would  be  the  side  of  one  containing  27  times  as  much  ? 
64  times  as  much  ?     125  times  as  much  ? 

Ans,  15,  20,  and  25  feet. 

5.  There   is  a  cubical   box,   measuring    1  foot  on  each 

side  ;  what  is  the  side  of  a  box  8  times  as  large  ?     27 

times  ?     64  times  ?  Ans,  2,  3,  and  4  feet. 

^  111,  Hence  we  see,  that  the  sides  of  cubes  are  as  the 
cube  roots  of  their  solid  contents^  and,  consequently,  their  con- 
tents are  as  the  cubes  of  their  sides.  The  same  proportion  is 
true  of  the  similar  sides^  or  of  the  diameters  of  all  solid  figures 
of  similar  forms. 

6.  If  a  ball,  weighing  4  pounds,  be  3  inches  in  diameter, 
what  will  be  the  diameter  of  a  ball  of  the  same  metal,  weigh- 
ing 32  pounds?  4  :  32  :  :  33  :  6-  ,  u4«5.  6  inches. 
\  7.  If  a  ball,  6  inches  in  diameter,  weigh  32  pounds,  what 
will  be  the  weight  of  a  ball  3  inches  in  diameter  ?  Ans.  4  lbs. 

8.  If  a  globe  of  silver,  1  inch  in  diameter,  be  worth  $  6, 
what  is  the  value  of  a  globe  1  for    in  diameter  ? 

Ans,  $110368. 
"0.  There  are  two  globes ;  one  of  them  is  1  foot  in  diame- 
ter, and  the  other  40  feet  in  diameter ;  how  many  of  the 
fmalier  globes  would  it  take  to  make  1  of  the  larger  ? 

Ans,  64000. 

10.  If  the  diameter  of  the  sun  is  112  times  as  much  as  the 
diameter  of  the  earth,  how  many  globes  like  the  earth  would 
it  take  to  make  one  as  large  as  the  siin  ?  Ana,  1404928. 

^,.  11.  If  the  planet  Saturn  is  1000  times  as  large  as  the 
earth,  and  the  earth  is  7900  miles  in  diameter,  what  is  the 
diameter  of  Saturn  ?  Ans,  79000  miles. 

12.  There  are  two  planets  of  equal  density;  the  diameter 
of  the  less  is  to  that  of  the  larger  as  2  to  9  ;  what  is  the  ra- 
tio of  their  solidities  ?  Ans.  ^S-h-  :  or.  as  8  to  729. 


Zit  ARITHMETICAL    PROGRESSION.       IT  111,112^. 

Note*  The  roots  of  most  powers  may  be  found  by  the 
square  and  cube  root  only  :  thus,  the  biquadrate,  or  4th  root, 
is  the  square  root  of  the  square  root ;  the  6th  root  is  the 
cube  root  of  the  square  root ;  the  8th  root  is  the  square  root 
of  the  4th  root ;  the  9th  root  is  the  cube  root  of  the  cube 
root,  &c.  Those  roots,  viz.  the  5tb,  7th,  11th,  &c.,  which 
are  not  resolvable  by  the  square  and  cube  roots,  seldom  oc- 
cur, and,  when  they  do,  the  work  is  most  easily  performed 
by  logarithms ;  for,  if  the  logarithm  of  any  number  be  divided 
by  the  index  of  the  root,  the  quotient  will  be  the  logarithm 
of  the  root  itself. 


ARXTiiniXiTSCAXi    PROaHESSIOSr. 

^  112.  Any  rank  or  series  of  numbers,  more  than  two, 
increasing  or  decreasing  by  a  constant  difference,  is  called  an 
Arithmetical  Series^  or  Progression, 

When  the  numbers  are  formed  by  a  continual  addition  of 
the  common  difference,  they  form  an  ascending  series  ;  but 
when  they  are  formed  by  a  continual  subtraction  of  the  com- 
mon difference,  they  form  a  descending  series, 

Th  «    \    ^'    ^)    "^j  ^)  11?  ^^j  1-5)  &:c.  is  an  ascending  series. 
^^'  (  15,  13,  11,  9,    7,    5,    3,  &c.  is  a  descending  series. 

The  numbeis  which  form  the  series  are  called  the  terms 
of  the  series.  The  first  and  laM  terms  are  the  extremes^  and 
the  other  term.s  are  called  the  means.  ^^ 

There  are  live  things  in  arithmetical  progression,  any  three 
of  which  being  given,  the  other  tioo  may  be  found  : — 

1st.  The/r5^  term. 

2d.  The  last  term. 

3d.  The  number  of  terms. 

4th.  The  common  difference, 

5th.  The  sum  of  all' the  terms. 

1.  A  man  bought  100  yards  of  cloth,  giving  4  cents  for  the 
first  yard,  7  cents  for  the  second,  10  cents  for  the  third^  and 
so  on,  with  a  common  difference  of  3  cents ;  what  was  the 
cost  of  the  last  yard  ? 

As  the  common  difference^  3,  is  added  to  every  yard  except 
the  last,  it  is  plain  the  last  yard  must  be  99  X  3,  =  297 
cents,  more  than  the  iirst  vard.  Ans.  301  cents. 


TT  112.  ARITHMETICAL    PROGRESSION.  22S 

Hence,  wheii  the  first  term^  the  common  difference^  and  the 
number  of  termSj  are  given^  to  find  the  last  term^ — Multiply  the 
number  of  terms,  less  1,  by  the  common  difference,  and  add 
the  first  term  to  the  product  for  the  last  term. 

2.  If  the  first  term  be  4,  the  common  difference  3,  and 
the  number  of  terms  100,  what  is  the  last  term  ?      Ans»  301. 

3.  There  are,  in  a  certain  triangular  field,  41  rows  of 
com ;  the  first  row,  in  1  corner,  is  a  single  hill,  the  second 
contains  3  hills,  and  so  on,  with  a  common  difference  of  2  ; 
what  is  the  number  of  hills  in  the  last  row.^      Ans.  81  hills. 

4.  A  man  puts  out  $1,  at  6  per  cent,  simple  interest, 
which,  in  1  year,  amounts  to  $  V06,  in  2  j^ears  to  $  142, 
and  so  on,  in  arithmetical  progression,  with  a  common  dif- 
ference of  $  ^06 ;  what  would  be  the  amount  in  40  years  ? 

Ans,  $3*40. 
Hence  we  see,  that  the  yearly  amounts  of  any  sum,  at 
simple  interest,  form  an  arithmetical  series,  of  which  the 
vrincipal  is  the  first  term,  the  last  amount  is  the  last  term,  the 
yearly  interest  is  the  cominon  difference^  and  the  number  of 
years  is  1  less  than  the  number  of  terms. 

5.  A  man  bought  100  yards  of  cloth  in  arithmetical  pro- 
gression ;  for  the  first  yard  he  gave  4  cents,  and  for  the  last 
.301  cents  ;  what  was  the  common  increase  of  the  price  on 
each  succeeding  yard  ? 

This  question  is  the  reverse  of  example  1 ;  therefore, 
301  — -  4  =  297,  and  297  ~  99  =i  3,  common  difference. 

j^jMence^   when  the  extremes  and  number  of  terms  are  given, 
^fl^d  the  common  difference, — Divide  the  difference  of  the 

extremes  by  tlie  number  of  terms,  less  1,  and  the  quotient 

will  be  the  common  difference. 

6.  If  the  extremes  be  5  and  605,  and  the  number  of  terms 
151,  what  is  the  common  difference?  Ans,  4. 

7.  If  a  man  puts  out  $  1,  at  simple  interest,  for  40  years, 
and  receives,  at  the  end  of  the  time,  $  3'40,  what  is  the 
rate  ? 

If  the  extremes  be  1  and  3^40,  and  the  number  of  terms 
41,  what  is  the  common  difference  ?  Ans,  *06. 

8.  A .  man  had  8  sons,  whose  ages  differed  alike ;  the 
youngest  was  10  years  old,  and  the  eldest  45 ;  what  was 
the  common  difference  of  their  acres  }  Ans.  5  years. 


224  ARITHMETIC AI^    PROGRESSION.  IF    112, 

9.  A  man  bought  100  yards  of  cloth  in  arithmetical  series; 
he  gave  4  cents  for  the  first  yard,  and  301  cents  for  the  last 
yard ;  what  was  the  average  price  per  yard,  and  what  was 
the  amount  of  the  whole  ? 

Since  the  price  of  each  succeeding  yard  increases  by  a  con- 
stant  excess^  it  is  plain,  the  average  price  is  as  much  less  than 
the  price  of  the  last  yard,  as  it  is  greater  than  the  price  of 
the  first  yard ;  therefore,  one  half  the  sum  of  the  first  and 
last  price  is  the  average  price. 

One  half  of  4  cts.  -\-  301  cts.  -.  152^  cts.  =  average  ^ 
price  ;   and  the  price,  152^  cts.  X  lOO^^i  15250  cts.=:  >  Ans. 
$152*50,  whole  cost.  ) 

Hence,  when  the  extremes  and  the  number  of  terms  are  given^ 
to  find  the  sum  of  all  the  terms^ — Multiply  J  the  sum  of  the  ex- 
tremes by  the  number  of  terms,  and  the  product  will  b« 
the  answer. 

10.  If  the  extremes  be  5  and  605,  and  the  number  of 
terms  151,  what  is  the  sum  of  the  series  ?  Aiis.  46055. 

11.  What  is  the  sum  of  the  first  100  numbers,  in  their 
natural  order,  that  is,  1,  2,  3,  4,  &:c.  ?  Ans,  5050. 

12.  How  many  times  does  a  common  clock  strike  in  12 
hours  ?  Ans.  78. 

13.  A  man  rents  a  house  for  $  50,  annually,  to  be  paid  at 
the  close  of  each  year;  what  will  the  rent  amount  to  in  20 
years,  allowing  6  per  cent.,  simple  interest,  for  the  use  of 
the  money  ? 

The  last  year's  rent  will  evidently  be  $  50  without  inteiest, 
the  last  but  one  will  be  the  amount  of  $  50  for  1  year , 
last  but  two  the   amount  of  $  50  for  2  years,  and  so  o: 
arithmetical  series,  to  the  first,  which  will  be  the  amount 
$50  for  19  years  ~  $107. 

If  the  first  term  be  50,  the  last  term  107,  and  the  number 
of  terms  20,  what  is  the  sum  of  the  series  ?        Ans.  $  1570, 

14.  What  is  the  amount  of  an  annual  pension  of  $  100, 
being  in  arrears,  that  is,  remaining  unpaid,  for  40  years, 
allowing  5  per  cent,  simple  interest  ?  Ans.  $  7900. 

15.  There  are,  in  a  certain  triangular  field,  41  rows  of 
com ;  the  first  row,  being  in  1  corner,  is  a  single  hill,  and 
the  last  row,  on  the  side  opposite,  contains  81  hills ;  how 
many  hills  of  corn  in  the  field  }  Ans,  1681  hills» 


nt  of 


II    112,   113.       GEOMETRICAL    PROGRESSION.  225 

16.  If  a  triangular  piece  of  land,  30  rods  in  length,  be  21 
rods  wide  at  one  end,  and  come  to  a  point  at  the  other,  wlic^ 
number  of  square  rods  does  it  contain  ?  Ans.  300, 

17.  A  debt  is  to  be  discharged  at  11  several  payments, 
in  arithmetical  series,  the  first  to  be  $5,  and  the  last  $75; 
what  is  the  whole  debt  ?  the  common  difference  be- 
tween the  several  payments  ? 

Ans.  whole  debt,  $  440 ;  common  difference,  $  7. 

18.  What  is  the  sum  of  the  series  1,  3,  5,  7,  9,  &c.,  to 
1001  ?  Ans.  251001. 

Note,  By  the  reverse  of  the  rule  under  ex.  5,  the  differ- 
ence of  the  extremes  1000,  divided  by  the  common  difference  2, 
gives  a  quotient,  which,  increased  by  1,  is  the  number  of 
terras  z=z  501. 

19.  What  is  the  sum  of  the  arithmetical  series  2,  2J-,  3, 
3^,  4,  4^,  &c.,  to  the  50th  term  inclusive  ?  Ans,  712^. 

20.  What  is  the  sum  of  the  decreasing  series  30,  29f ,  29^, 
29,  28f ,  &c.,  down  to  0  ? 

Note,      30  -i-  ^  4-  1  =  91,  number  of  terms,      xins,  1365. 

QUESTIONS. 

1.  What  is  an  arithmetical  progression?     2.  When,  is  the 

series  called  ascending  1    3. •  ^vhen  descending!    4.  What 

are  the  numbers,  forming  the  progression,  called  ?  5.  What 
are  the  first  and  last  terms  called  ?  6.  What  are  the  other 
terms  called?  7.  When  the  first  ?er??i,  common  difference, 
aiid  number  of  terms,  are  given,  how  d^^ou  find  the  last 
term  ?     8.  How  may  arithmetical  progression  be  applied  to 

imple  interest?  9.  When  the  extremes  and  number  of 
is  are  given,  how  do  you  find  the  common  difference  ? 

9.  how  do  you  find  the  sum  of  all  the  terms  ? 


^. 


TF  113-  Any  series  of  numbers,  continually  increasing  by 
a  constant  multiplier,  or  decreasing  by  a  constant  divisor,  ig 
called  a  Geometrical  Progression.  Thus,  1,  2,  4,  8,  16,  &c. 
IS  an  increasing  geometrical  series,  and  8,  4,  2,  1,  ^,  jr,  ^fcc* 
is  a  decreasing  geometrical  series. 


tt6  GEOMETRICAL   PROGRESSION.  IT  11$^ 

As  in  aritlimetical,  so  also  in  geometrical  progression^ 
there  are  five  things,  any  three  of  which  being  given,  the 
other  two  may  be  found  : — 

1st  The /r5/ term. 
2d.  The  last  term. 
3d.  The  number  of  terms. 
4th.  The  ratio. 

5tii.  The  sum  of  all  the  terms. 

The  ratio  is  the  multiplier  or  divisoTy  by  which  the  series  i# 
formed. 

1.  A  man  bought  a  piece  of  silk,  measuring  17  yards,  and, 
by  agreement,  was  to  give  what  the  last  yard  would  come 
to,  reckoning  3  cents  for  the  first  yard,  6  cents  for  the  second, 
and  so  on,  doubling  the  price  to  the  last;  what  did  the  piece 
of  silk  cost  him  ? 

3X2X2X2X2X2X2X2X2X2X2X2X2 
X2X2X2X2=:  196608  cents,  =  $  1966*08,  Answer, 

In  examining  the  process  by  which  the  last  term  (196608) 
has  been  obtained,  we  see,  that  it  is  a  product,  of  which  the 
ratio  (2)  is  sixteen  times  a  factor,  that  is,  one  time  less  than 
the  number  of  terms.  The  last  term,  then,  is  the  sixteenth 
power  of  the  ratio^  (2,)  multiplied  by  ih^  first  term  (3.) 

Now,  to  raise  2  to  the  16th  power,  we  need  not  produce 
all  the  intermediate  powers  ;  for2^i=i2X2X2X2z=:16, 
is  a  product  of  whi<,'h  the  ratio  2  is  4  times  a  factor ;  now, 
if  16  be  multiplied  by  16,  the  product,  256,  evidently  con- 
tains the  same  factor  (2)  4  times  -(-  4  times,  z=.  8  times; 
and  256  X  256  =:  65536,  a  product  of  which  the  ratio  (2) 
is  8  times  -{-  8  times,  zn  16  times,  factor ;  it  is,  therefore, 
the  16th  power  of  2,  and,  multiplied  by  3,  the  first  term, 
gives  196608,  the  last  term,  as  before.     Hence, 

When  the  first  term,  ratio,  and  number  of  terms,  are  given, 
to  find  the  last  term, — 

I.  Write  down  a  few  leading  powers  of  the  ratio  with 
their  indices  over  them. 

II.  Add  together  the  most  convenient  indices,  to  make  an 
index  less  by  one  than  the  number  of  the  term  sought. 

III.  Multiply  together  iha  powers  belonging  to  those  t7>- 
dices,  and  their  product,  multiplied  by  the  first  term^  will  b« 
the  term  sought 


f  113.  GEOMETRICAL   PROGRESSION.  227 

2.  If  the  first  term  be  6,  and  the  ratio  3,  what  is  the  8th 
term  ? 

Powers  of  the  ratio,  with  ^\    l    0^7  v  ft i  ~  sil^  v  ^  fir«f 
their  indices  over  them.  )  ^'  /'  ^^'  X,  ^^  ~  7^^  X  ^  ^^^ 
(      term,  ziz  10935,  Answer, 

3.  A  man  plants  4  kernels  of  corn,  which,  at  harvest, 
produce  32  kernels  ;  these  he  plants  the  second  year ;  now, 
supposing  the  annual  increase  to  continue  8  fold,  what 
would  be  the  produce  of  the  16th  year,  allowing  1000  ker- 
nels to  a  pint  ?  Am,  2199023255'552  bushels. 

4.  Suppose  a  man  had  put  out  one  cent  at  compound  in- 
terest in  1620,  what  would  have  been  the  amount  in  1824, 
allowing  it  to  double  once  in  12  years  ? 

217  —  131072.  ^715.    $1310*72. 

5.  A  man  bought  4  yards  of  cloth,  giving  2  cents  for  the 
first  yard,  6  cents  for  the  second,  and  so  on,  in  b  fold  ra- 
tio ;  whai  did  the  whole  cost  him  ? 

2  +  6  +  18  +  54  =  80  cents.  Ans.  80  cents. 

In  a  long  series,  the  process  of  adding  in  this  manncy 
would  be  tedious.  Let  us  try,  therefore,  to  devise  some 
shorter  method  of  coming  to  the  same  result.  If  all  the 
terms,  excepting  the  last^  viz.  2  -|- 6  +  18,  be  multiplied  by 
the  ratio,  3,  the  product  will  be  the  series  6  -f-  18-f-54 
subtracting  the  former  series  from  the  latter^  we  have,  for  the 
remainder,  54  —  2,  that  is,  the  last  term,  less  the  first  term^ 
which  is  evidently  as  many  times  the  first  series  (2  -f-  6  -j-  18) 
as  is  expressed  by  the  ratio,  less  1 :  hence,  if  we  dlmde  the 
difference  of  the  extremes  (54  —  2)  by  the  ratio,  less  1, 
(3  —  1,)  the  quotient  will  be  the  sum  of  all  the  terms,  ex" 
cepting  the  last^  and,  adding  the  last  term,  we  shall  have  the 
xohole  amount.  Thus,  54  —  2  z=  52,  and  3  —  1  z=  2 ;  then, 
52  -i-2  :z=:  26,  and  54  added,  makes  80,  Answer,  as  before. 

Hence,  when  the  extremes  and  ratio  are  given,  to  fmd  the 
gum  of  the  series, — Divide  the  difference  of  the  extremes  by  the? 
ratio,  less  1,  and  the  quotient,  increased  by  the  greater  terviy 
will  be  the  answer. 

6.  If  the  extremes  be  4  and  131072,  and  the  ratio  8, 
what  is  the  whole  amount  of  the  series  ? 

131072  —  4    ,    131072  =  149796  Anruf^r. 
8—1         ' 


&28  GEOMETRICAL    PROGRESSION.  IT  113. 

7.  What  is  the  sum  of  the  descending  series  3,  1,  ^,  J, 
2^,  &c.,  extended  to  infinity? 

It  is  evident  the  last  term  must  become  0,  or  ind^^finitely 
near  to  nothing ;  therefore,  the  extremes  are  3  and  0,  and 
the  ratio  3.  Ans,  4^. 

8.  What  is  the  value  of  the  infinite  series  1  +  i  +  iV  + 
^V,  &c.  ?  Alls.  4. 

9.  What  is  the  value  of  the  infinite  series,  -J^y  +  tott  + 
T7T0U  +  T^iTTU?  ^^-y  ^^3  what  is  the  same,  the  decimal 
41111,  &c.,  continually  repeated?  Ans.  -J. 

10.  What  is  the  value  of  the  infinite  series,  y-^-^y  -)-  y^jg^nj) 
&c.,  descending  by  the  ratio  100,  or,  which  is  the  same,  the 
repeating  decimal  ^020202,  &c.  ?  -4725.  -^^, 

11.  A  gentleman,  whose  daughter  was  married  on  a  new 
year's  day,  gave  her  a  dollar,  promising  to  triple  it  on  the 
first  day  of  each  month  in  the  year ;  to  how  much  did  her 
portion  amount? 

Here,  before  finding  the  amount  of  the  series,  we  must 
find  the  last  term^  as  directed  in  the  rule  after  ex.  1. 

Ans.    $265^720 

The  two  processes  of  finding  the  last  term^  and  the  omount, 
may,  however,  be  conveniently  reduced  to  o?ie,  thus  : — 

JVhen  the  first  terrn^  the  ratio,  and  the  number  of  terms,  are 
given,  to  find  the  sum  or  amount  of  the  senes, — Raise  the  ratio 
to  a  power  whose  index  is  equal  to  the  number  of  terms,  from 
which  subtract  1 ;  divide  the  remainder  by  the  ratio,  less  1, 
and  the  quotient,  multiplied  by  the  first  term,  will  be  the 
answer. 

Applying  this  ^ule  to  the  last  example,  3^2  —  531441, .a»d 
531441  —  1  ^  ^  __  265720.  Ajis.    $  265^720,  as  hdofi 

o 1 

12.  A  man  agrees  to  serve  a  farmer  40  years  without  any 
other  reward  than  1  kernel  of  corn  for  the  first  year,  10  for 
the  second  year,  and  so  on,  in  10  fold  ratio,  till  the  end  of 
the  time  ;  what  will  be  the  amount  of  his  wages,  allowing 
1000  kernels  to  a  pint,  and  supposing  he  sells  his  corn  at  50 
cents  per  bushel  ? 

10<to_i  ^.  T  __  J  1,111,111,111,111,111,111,111,111, 
10~-i~^     "~  (      111,111,111,111,111  kernels. 
Ans.    $  8,680,555,555,655,555,555,555,556,555,656,665 
^555/^V 


^  113,  114.       GEOMETllIt'At.   PROGRESSION.  229 

13.  A  gentleman,  dying,  left  his  estate  to  his  5  sons,  to 
the  youngest  $1000,  to  the  second  $1500,  and  ordered, 
that  each  son  should  exceed  the  younger  Ij  the  ratio  of  l^ ; 
what  was  the  amount  of  the  estate  ? 

Note,  Before  finding  the  power  of  the  ratio  1^,  it  may 
he  reduced  to  an  improper  fraction  =:  f ,  or  to  a  decimal,  1*6. 

^^""^  X  1000  ==  $131874-;  or,  l-51izi    X  1000  =: 
|.  — 1  ^  ^        '  1*5  — 1 

$13187*50,  Answer. 

Compound  Interest  by  Progression. 

IT  114.  1.  Wliat  is  the  amount  of  $4,  for  5  years,  at  6 
per  cent,  compound  interest  ? 

We  have  seen,  (IT  92,)  that  compound  interest  is  that, 
which  arises  from  adding  the  interest  to  the  principal  at  the 
close  of  each  year,  and,  for  the  next  year,  casting  the  inter- 
est on  that  amount^  and  so  on.  The  amount  of  $  1  for  1 
year  is  $  1*06  ;  if  the  principal^  therefore,  be  multiplied  by 
1*06,  the  product  will  be  its  amount  for  1  year  ;  this  amount^ 
BiultipJied  by  1*06,  will  give  the  amount  (compoimd  inter- 
est) for  2  years;  and  this  second  amount^  multiplied  by  1*06, 
will  give  the  amount  for  3  years  ;  and  so  on.  Hence, 
the  several  amounts^  arising  from  any  sum  at  compound  in- 
terest, form  a  geometrical  series^  of  which  the  princi]}al  is  the 
first  term;  the  amount  of  $  I  or  1  £ .^  &c.,  at  the  giveii  rate 
per  cent, J  is  the  ratio  ;  the  time^  in  years j  is  1  less  than  the 
number  of  tenns ;  and  the  last  amount  is  the  last  term. 

The  last  question  may  be  rc?'-lved  into  this : — If  the  first 
term  be  4,  the  number  of  terms  6,  and  the  ratio  1*06, 
what  is  the  last  term  ? 

r065=:l*338,and  1*338X4=  $5*352+.      Ans.    $5*352. 

Note  1.  The  powers  of  the  amo7:ints  of  $  1,  at  5  and  at  6 
per  cent.,  may  be  taken  from  the  table,  under  IT  91.  Thus, 
opposite  5  years,  under  6  percent,  you  find  1*338,  &c. 

Note  2.  The  seveial  processes  may  be  conveniently  exhi- 
bited by  the  use  of  letters ;  thus : — 
Let  P.  represent  the  Principal. 

R the  Ratio,  or  the  amount  of  $  1^  &c.  for  1  year. 

••...  T the  Time,  in  years, 

....  A the  Amount. 

When  two  or  more  letters  are  joined  together^  like  a  worrl, 


230  GEOMETRICAL    PROGRESSION.  IF   114. 

they  are  to  be  multiplied  together.  Thus  PR.  implies,  that 
the  principal  is  to  be  multiplied  by  the  ratio.  When  one 
letter  is  placed  chove  another,  like  the  index  of  a  power,  the 
jlrst  is  to  be  raised  to  a  power,  whose  index  is  denoted  by  tne 
second.  Thus  R'^-  implies,  that  the  ratio  is  to  be  raised  to 
a  power,  whose  index  shall  be  equal  to  the  time,  that  is,  the 
number  of  years. 

2.  What  is  the  amount  of  40  dollars  for  11  years,  at  5  per 
cent,  compound  interest? 

}V^'  X  P.  —  A. ;  therefore,  V05^^  X  40  z=: 68'4. 

Ans.    $68^40. 

3.  What  is  the  amount  of  $6  for  4  years,  at  10  per  cent. 
compound  interest  ?  Ans.    $8'784^^. 

4.  If  the  amount  of  a  certain  sum  for  5  years,  at  6  per 
cent,  compound  interest,  be  $  5^352,  what  is  that  sum,  or 
principal  ? 

If  the  number  of  terms  be  6,  the  ratio  1^06,  and  the  last 
term  5^352,  what  is  the  first  term  ? 

This  question  is  the  reverse  of  the  last;  therefore, 

6.  What  principal,  at  10  per  cent,  compound  interest,  will 
tmount,  in  4  years,  to  $  S'7S46  ?  Ans.    $  6. 

6.  What  is  the  present  worth  of  $68^40,  due  11  years 
nence,  discounting  at  the  rate  of  5  per  cent,  compound  in- 
terest? Ans.    $40. 

7.  At  what  rate  per  cent  will  $  6  amount  to  $  8'7846  in 
4  years  ? 

If  the  first  term  be  6,  the  last  term  8'7S46,  and  the  num 

ber  of  terms  5,  what  is  the  ratio  ? 

A.     -t>T   IT,  4-     8^7846 

_  =  K^-,thatis,  — - —  3-.  1^4641  z=:  the  4th  power  of 

the  ratio  ;  and  then,  by  extracting  the  4th  root,  we  obtain 
140  ^r  the  ratio.  Aiis.  10  per  cent. 

8.  In  what  time  will  $6  amount  t<»  $8'7846,  at  10  per 
cent  compound  interest  ? 

p^  =  RT.,  that  is,  — g—  =  1^4641  =r  140^- ;  therefore, 

if  we  divide  1'4641  by  140,  and  tlien  divide  the  quotient 
thence  arising  by  140,  and  so  on,  till  we  obtain  a  quotient 
that  will  not  contain  140,  the  nziiider  of  tliese  divisions  will 
b«  the  number  of  years,  Ans,  4  year*. 


*r   114,  115.      GEOMETRICAL    PROGRESSION.  2S1 

9.  At  5  per  cent,  compound  interest,  in  what  time  will 
$  40  amount  to  $  68'40  ? 

Having  found  the  power  of  the  ratio  1*05,  as  heforc,  which 
is  1*71,  you  may  look  for  this  number  in  the  table^  under 
the  given  rate,  5  per  cent.,  and  against  it  you  will  find  the 
number  of  years.  Ans.  11  years. 

10.  At  6  per  cent,  compound  interest,  in  what  time  will 
$  4  amount  to  $  5*352  ?  Ans.  5  years. 


Annuities  at  Compound  Interest, 

IT  115,  It  may  not  be  amiss,  in  this  place,  briefly  to  show 
the  application  of  compound  interest,  in  computing  the 
amount  and  present  worth  of  annuities. 

An  Annuity  is  a  sum  payable  at  regular  periods^  of  one 
year  each,  either  for  a  certain  number  of  years,  or  during  the 
life  of  the  pensioner,  or  forever. 

When  annuities,  rents,  &c.  are  not  paid  at  the  time  they 
become  due,  they  are  said  to  be  in  arrears. 

The  sum  of  all  the  annuities,  rents,  &c.  remaining  un- 
paid, together  with  the  interest  on  each,  for  the  time  they 
have  remained  due,  is  called  the  amount, 

1.  What  is  the  amount  of  an  annual  pension  of  $  160, 
which  has  remained  unpaid  4  years,  allowing  6  per  cent, 
compound  interest? 

The  te/  yearns  pension  will  be  $  100,  without  interest ; 
the  last  but  one  will  be  the  amount  of  $  100  for  1  year;  the 
last  but  two  the  amount  (compound  interest)  of  $  100  for 
2  years,  and  so  on ;  and  the  sum  of  these  several  amounts 
will  be  the  answer.  We  have  then  a  series  of  amounts,  (hat 
is,  di  geometrical  series,  (IT  114,)  to  find  the  sum  of  all  the 
terms. 

If  the  first  term  be  100,  the  number  of  terms  4,  and  the 
ratio   1*06,  what  is  the  sum  of  all  the  terms  ? 

Consult  the  rule,  under  IT  113,  ex.  11. 

1«06*— -1 

—^ X  100  ~  437*45.  Ans.  $437*45. 

Hence,  when  the  annuity,  the  time,  and  rate  per  cent,  are 
given,  to  find  the  amount, — Raise  tlie  ratio  (the  amount  of 


232  GEO:VI£TRfCAL    PROGRESSION.       IF  115,116. 

$  1,  &c.  for  1  year)  to  a  power  denoted  by  the  number  of 
years;  from  this  power  subtract  1 ;  then  divide  the  remaiiH 
der  by  the  ratio,  less  1,  and  the  quotient,  multiplied  by 
the  annuity,  will  be  the  amount. 

Note,  The  powers  of  the  amounts,  at  5  and  6  per  cent, 
up  to  the  24th,  may  be  taken  from  the  table^  under  1\  91. 

2.  What  is  the  amount  of  an  annuity  of  $  50,  it  being  in 
arreais  20  years,  allowing  5  per  cent,  compound  interest? 

Ans.  $]653<29. 

3.  If  the  annual  rent  of  a  house,  which  is  $  150,  be  in 
arrears  4  years,  what  is  the  amount,  allowing  10  per  cent. 
compound  interest  ?  Am,  $69645, 

4.  To  how  much  would  a  salary  of  $600  per  annum 
amount  in   14  years,  the  money  being  improved  at  6  per 

cent,  compound  interest?      in  10  years?     in  20 

years  ?     in  22  years  ?     in  24  years  ? 

Am,  to  the  last,  $  25407'75. 

IT  116.  If  the  annuity  is  paid  in  advance,  or  if  it  be 
bought  at  the  beginning  of  the  first  year,  the  sum  which 
ought  to  be  given  for  it  is  called  the  present  worth, 

5.  What  is  the  present  worth  of  an  annual  pension  of 
$  100,  to  continue  4  years,  allowing  6  per  cent,  compeund 
interest  ? 

The  present  worth  is,  evidently,  a  sum  which,  at  6  per 
cent,  compound  interest,  would,  in  4  years,  produce  an  amount 
equal  to  the  ainoimt  of  the  annuity  in  arrears  the  same  time. 

By  the  last  rule^  we  find  the  amount  zz:  $437'45,  and  by 
the  directions  under  ^  114,  ex.  4,  we  find  the  present  w^orth 
=  $346'51.  Am.  $346^51. 

Hence,  to  find  the  present  worth  of  any  annuity, — ^First 
find  its  amount  in  arrears  for  the  whole  time ;  this  amount j 
divided  by  that  power  of  the  ratio  denoted  by  the  number  of 
years,  will  give  the  present  worth, 

6.  What  is  the  present  worth  of  an  annual  salary  of  $  100 
to  continue  20  years,  allowing  5  per  cent  ?  Ans.  $  1246'22. 


If  116. 


GEOMETRICAL    PROGRESSION. 


233 


The  operations  under  this  rule  being  somewhat  tedious, 
we  subjoin  a 

TABL.E, 

Showing  the  present  worth  of  $  1,  or  1  £,  annuity,  at  5  and 
6  per  cent,  compound  interest,  for  any  number  of  years 
from  1  to  34. 


Yea«. 

1 

2 
3 

4 
5 
6 
7 
8 
9 

10 

11 

12 

13 

14 

15 

16 

17 


5  per  cent. 

0'95238 
1^85941 
2^72325 
3'54595 
4^32948 
5^07569 
5'78637 
6'46321 
74  0782 
7'72173 
8'30641 
8^86325 
9^39357 
9'89864 
10'37966 
10^83777 
11^27407 


6  per  cent.  | 

Yeans. 

0'94339 

18 

1 '83339 

19 

2^67301 

20 

3'4651 

21 

4^21236 

22 

4'91732 

23 

5^58238 

24 

6^20979 

25 

6'80169 

26 

r36008 

27 

7*88687 

28 

8*38384 

29 

8'85268 

30 

9*29498 

31 

9*71225 

32 

10*10589 

33 

10*47726 

34 

5  per  cent. 

11*68958 

12*08532 

12*46221 

12*82115 

13*163 

13*48807 

13*79864 

14*09394 

14*37518 

14*64303 

14*89813 

15*14107 

15*37245 

15*59281 

15*80268 

16*00255 

16*1929 


6  per  cent. 

10*8276 

11*15811 

11*46992 

11*76407 

12*04158 

12*30338 

12*55035 

12*78335 

13*00316 

13*21053 

13*40616 

13*59072 

13*76483 

13*92908 

14*08398 

14*22917 

14*36613 


It  is  evident,  that  the  present  worth  of  $  2  annuity  in  2 
times  as  much  as  that  of  ^  1 ;  the  present  worth  of  $  3  will 
be  3  times  as  much,  &c.  Hence,  to  find  the  present  worth 
of  any  ammity^  at  5  or  6  per  cent.^ — Find,  in  this  table,  the 
present  worth  of  $  1  annuity,  and  multiply  it  by  tlie  given 
annuity^  and  the  product  will  be  the  present  worth, 

7.  What  ready  money  will  purchase  an  annuity  of  $  150, 
to  continue  30  years,  at  5  per  cent,  compound  interest? 

The  present  worth  of  $  1  annuity,  by  the  table,  for 
30  years,  is  $  15*37245  ;  therefore,  15*37245  X  150  :=z 
$  2305*367,  Ans. 

8.  What  is  the  present  worth  of  a  yearly  pension  of  $  40, 
at  6  per  cent,  compoaud  interest? 
to  continue  15  years  ?     —  20 

34  years  ? 


to 


contmue  10  years, 

at  5  per  cent.  ?     — 

years  ?    25  years  ? 


Ans.  to  last,  $  647^716. 


234  GEOMETRICAL   PROGRESSrblS?.  It  116. 

When  annuities  do  not  commence  till  a  certain  period  of 
time  has  elapsed,  or  till  some  particular  event  has  taken 
place,  they  are  said  to  be  in  reversion, 

9.  What  is  the  present  worth  of  $  100  annuity,  to  be 
continued  4  years,  but  not  to  commence  till  2  years  hence, 
allowing  6  per  cent,  compound  interest  ? 

The  present  worth  is  evidently  a  sum  which,  at  6  per 
cent,  compound  interest,  would  in  2  years  produce  an  amount 
equal  to  the  present  worth  of  the  annuity,  were  it  to  commence 
immediateli/.  By  the  last  rule,  we  find  the  present  worth  of 
tbe  annuity,  to  commence  immediately,  to  be  $346 '51,  and, 
by  directions  under  IT  114,  ex.  4,  we  find  the  present  worth 
of  $346'51  for  2  years,  to  be  $30S'393.      Am.  $308'393. 

Hence,  to  find  the  present  worth  of  any  annuity  taken  in 
reversion,  at  compound  interest, — First,  find  the  present  worth, 
to  commence  immediately,  and  this  sum,  divided  by  the  power 
of  the  ratio,  denoted  by  the  time  in  reversion,  will  give  the 
answer. 

10.  What  ready  money  will  purchase  the  reversion  of  a 
ease  of  $  60  per  annum,  to  continue  6  years,  but  not  to  com- 
mence till  tlie  end  of  3  years,  allowing  6  percent,  compound 
interest  to  the  purchaser  ? 

The  present  worth,  to  commence  immediately,  we  find  to 

be,  $  295^039,  and  -r,;r^  ==  247^72.  Am,  $  247*72. 

1  06*^ 

It  is  plain,  the  same  result  will  be  obtained  by  finding  the 
present  v/orth  of  the  annuity,  to  commence  immediately, 
and  lo  continue  to  the  end  of  the  time,  that  is,  3  -|-  6  =  9 
years,  and  then  subtracting  from  this  sum  the  present  worth 
of  the  annuity,  continuing  for  the  time  of  reversion,  3  years. 
Or,  wc  may  find  the  present  worth  of  $  1  for  the  two  times 
by  the  table,  and  multiply  their  difference  by  the  given  an- 
nuity.   Thus,  by  the  table. 

The  whole  time,         9  years,  m  6*80169 
The  time  in  reversion,  3  years,  =  2*67301 

Diff*erence,  =  4*12868 
60 


$247*72080    Ans, 
11.  What  is  the  present  worth  of  a  lease  of  $  100  to  con- 
tinue 20  years,  but  not  to  commence  till  the  end  of  4  years, 


1r  lit.  GEOMETRICAL    PROGRESSION.  235 

allowing  5  per  cent.  ?     what,  if  it  be  6  years  in  rever- 
sion ?     8  years  ?     10  years  ?     14  years  ? 

Am,  to  last,  $  629^426. 

TT 117.  12.  What  is  tb^  worth  of  a  freehold  estate,  of 
which  the  yearly  rent  is  $  60,  allowing  to  the  purchaser 
6  per  cent.  ? 

In  this  case,  the  annuity  continues  forever,  and  the  estate 
is  evidently  worth  a  sum,  of  which  the  yearly  interest  is  equal 
to  the  yearly  rent  of  the  estate.  The  principal  multiplied  by 
the  rate  gives  the  interest;  therefore,  the  interest  dimded 
by  the  rate  will  give  the  principal ;   60  -r-«06  nz  1000. 

A?is.  $1000. 

Hence,  to  find  the  present  worth  of  an  annuity,  continuing 
forever, — Divide  the  annuity  by  the  rate  per  cent.,  and  the 
quotient  will  be  the  present  worth. 

Note,  The  worth  will  be  the  same,  whether  we  reckon 
simple  or  compound  interest ;  for,  since  a  yearns  interest  of  the 
mice  is  the  annuity,  the  profits  arising  from  that  price  can 
neither  be  more  nor  less  than  the  profits  arising  from  the  an-- 
nuity,  whether  they  be  employed  at  simple  or  compound  in- 
terest. 

13.  What  is  the  worth  of  $  100  annuity,  to  continue  for- 
ever, allowing  to  the  purchaser  4  per  cent.  ?     allowing 

6  percent.?    8  percent.?    10  percent.  ?    15 

per  cent.  ?     20  per  cent.  ?  Ans,  to  last,  $  500. 

14.  Suppose  a  freehold  estate  of  $  60  per  annum,  to  com- 
mence 2  years  hence,  be  put  on  sale ;  what  is  its  value,  al- 
lowing the  purchaser  6  per  cent.  ? 

Its  present  worth  is  a  sum  which,  at  6  per  cent,  compound 
interest,  would,  in  2  years,  produce  an  amount  equal  to  the 
worth  of  the  estate  if  entered  on  immediately, 

— —  z=  $  1000  ==  the  worth,  if  entered   on   immediately, 

tfc  1000 
and  — jTQg^  =  $  889^996,  the  present  worth. 

The  same  result  may  be  obtained  by  subtracting  from  the 
worth  of  the  estate,  to  commence  immediately,  the  present  worth 
of  the  annuity  60, /or  2  years,  the  time  o/ reversion.  Thus, 
by  the  table,  the  present  worth  of  $  1  for  2  years  is  1^83339 
X  60  ==  110*0034  m  present  worth  of  $60  for  2  years, 
and  $  1000  --  $  110'o634  =:  $889*9966,  Ans,  as  before. 


£36  GE03IETRICAL    PROGRESSION.  IT  117. 

15.  What  is  the  present  worth  of  a  perpetual  annuity  of 
$  100,  to  commence  6  years  hence,  allowing  the  purchaser 
5  per  cent,  compound  interest  ? what,  if  8  years  in  re- 
version ?     10  years  ?     4  years  ?     15  years  ? 

30  years  ?  Ans,  to  last^  $  462'755. 

The  foregoing  examples,  in  compound  interest,  have  been 
confined  to  yearly  payments ;  if  the  payments  are  half  year- 
ly, we  may  take  half  the  principal  or  annuity^  half  the  rate  per 
cent.  J  and  twice  the  number  of  years ^  and  work  as  before,  and 
so  for  any  other  part  of  a  year, 

QUESTIONS. 

1.  What  is  a  geometrical  progression  or  series  ?  2.  What 
is  the  ratio  ?  3.  When  the  first  term,  the  ratio,  and  the  num- 
ber of  terms,  are  given,  how  do  you  find  ^^  last  term! 
4.  When  the  extremes  and  ratio  are  given,  how  do  you  find 
the  sum  of  all  the  terms  ?  5.  When  the  first  term,  the  ratio, 
and  the  number  of  ternis,  are  given,  how  do  you  find  the 
amount  of  the  series  ?  6.  When  the  ratio  is  a  fraction^  how 
do  you  proceed  ?  7.  What  is  compound  interest  ?  8.  How 
does  it  appear  that  the  amountSj  arising  by  compound  in- 
terest, form  a  geometrical  series  ?     9.  What  is  the  ratio^  in 

compound  interest  ?     the  number  of  teriJis  ?     the 

first  term?     the  lust  term?     10.  When  the  rate,  the 

time,  and  the  principal,  are  given,  how  do  you  find  the 
amount  ?  11.  "When  A.  R.  and  T.  are  given,  how  do  you  find 
P.  ?  12.  When  A.  P.  and  T.  are  given,  how  do  you  find  R.  ? 
13.  When  A,  P.  and  R.  are  given,  how  do  you  find  T.  ?  14. 
What  is  an  aminity  ?  15.  When  are  annuities  said  to  be  in  ar- 
rears ?  16.  What  is  the  amount  ?  17.  In  a  geometrical  series, 
to  what  is  the  araoimt  of  an  annuity  equivalent  ?  18.  How  do 
you  find  the  amount  of  an  annuity,  at  compound  interest? 

19.  What  is  the  present  worth  of  an  annuity  ?  how  com- 
puted at  compound  interest  ?     how  found  by  the  table  ? 

20.  What  is  understood  by  the  term  reversion  ?  21.  How 
do  you  find  the  present  worth  of  an  annuity,  taken  in  rever^ 

sion  ?    by  the  table  ?    22.  How  do  you  find  the  present 

worth  of  a  freehold  estate,  or  a  perpetual  annuity  ?     the 

same  taken  in  reversion  1    by  the  table  ? 


It  118,  119.       MISCELLANEOUS    EXAMPLES.  SSt 


PJESRIKEUTATZON. 

IT  118.  Permutation  is  the  method  of  finding  how  many 
different  ways  the  order  of  any  number  of  things  may  he 
varied  or  changed. 

1.  Four  gentlemen  agreed  to  dine  together  so  long  as 
they  could  sit,  every  day,  in  a  different  order  or  position ; 
how  many  days  did  they  dine  together  ? 

Had  there  been  but  two  of  them,  a  and  6,  they  could  sit 
only  in  2  times  1  (1  X  2z=:2)  different  positions,  thus, 
a  6,  and  b  a.  Had  there  been  three^  a,  b,  and  c,  they  could 
sit  in  1  X  2  X  3  zz:  6  different  positions ;  for,  beginning  the 
order  with  a,  there  will  be  2  positions,  viz.  a  b  Cj  and  ac  b  ; 
next,  beginning  with  by  there  will  be  2  positions,  b  a  c,  and 
b  c  a;  lastly,  beginning  with  c,  we  have  c  a  bj  and  c  b  a, 
that  is,  in  all,  1  X  2  X  3  =  6  different  positions.  In  the 
same  manner,  if  there  be  fouvy  the  different  positions  will 
De  1  X  2  X  3  X  4  z=:  24.  A?is,  24  days. 

Hence,  to  find  the  number  of  different  changes  or  permu- 
tationSj  of  which  any  number  of  different  things  are  capable^ — 
Multiply  continually  together  all  the  terms  of  the  natural 
series  of  numbers,  from  1  up  to  the  given  number,  and  the 
last  product  will  be  the  answer. 

2.  How  many  variations  may  there  be  in  the  position  of 
the  nine  digits  ?  Ans,  362880. 

3.  A  man  bought  25  cows,  agreeing  to  pay  for  them  1 
cent  for  every  different  order  in  which  they  could  all  be 
placed  ;  how  much  did  the  cows  cost  him  ? 

Ans.  $  155112100433309859840000. 

4.  Christ  Church,  in  Boston,  has  8  bells  ;  how  many 
changes  may  be  rung  upon  them  ?  Ans.  40320. 


TaiSCHI^I^AMlSOUS  SXAmTIiES. 


IT  119.     1.  4  +  6  X  7  —  1  =z60. 

A  line,  or  vinculum^  drawn  over  several  numbers,  signifies, 
that  the  numbers  under  it  are  to  be  taken  jointly,  or  as  one 
whole  number. 


jS38  miscellaneous  examples.  If  119. 


2.  9  — 8  +  4X8  +  4  —  6  =  how  many  ?  Ans.  30. 


3.  7  +  4  —  2  +  3  +  40X5  =  how  many  ?      Ans.  230. 


4.  3  +  6  — 2X4--2  __  ^^^  ^^^y  P  ^^    3^ 

2X2 
6.  There  are  two  numbers ;  the  greater  is  25  times  78, 
and  their  difference  is  9  times  15 ;  their  sum  and  product 
are  required. 

Ans,  3765  is  their  sum ;  3539250  their  product 

6.  What  is  the  difference  between  thrice  five  and  thirty, 
and  thrice  thirty-five  ?        35  X  3  —  5  X  3  +  30  =  60,  Ans. 

7.  What  is  the  diff*erence  between  six  dozen  dozen,  and 
half  a  dozen  dozen  ?  Ans.  792. 

8.  What  number  divided  by  7  will  make  6488  ? 

9.  What  number  multiplied  by  6  will  make  2058  ? 

10.  A  gentleman  went  to  sea  at  17  years  of  age;  8  years 
after  he  had  a  son  born,  who  died  at  the  age  of  35 ;  after 
whom  the  father  lived  twice  20  years ;  how  old  was  the 
father  at  his  death  ?  Ans.  100  years* 

11.  What  number  is  that,  which  being  multiplied  by  15 
the  product  will  he  ^?  J  -i-  15  =  g^y,  Ans. 

12.  What  decimal  is  that,  which  keing  multiplied  by  15, 
the  product  will  be  '75  ?  '75  -r- 15  =  '05,  Ans. 

13.  What  is  the  decimal  equivalent  to  ^  ? 

Ans.  '0285714. 

14.  What  fraction  is  that,  to  which  if  you  add  f ,  the  sum 
will  be  f  ?  Ans.  i^. 

15.  What  number  is  that,  from  which  if  you  take  f,  the 
remainder  will  be  ^  ?  Ans.  f  §. 

16.  What  number  is  that,  wiiich  being  divided  by  f,  the 
quotient  will  be  21  ?  Ans.   loj. 

17.  What  number  is  that,  which  multiplied  by  f  pro- 
duces ^  ?  Am.  |. 

18.  What  number  is  that,  from  which  if  you  take  |  of 
itself,  the  remainder  will  be  12  ?  Ans.  20. 

19.  What  number  is  that,  to  which  if  you  add  f  of  J-  of 
Itself,  the  whole  will  be  20  ?  Ais.  12. 

20.^  What  number  is  that,  of  which  9  is  the  §  part  ? 

Ans.  13^. 

21.  A  farmer  carried  a  load  af  produce  to  maiket:  he 
sold  780  lbs.  of  pork,  at  6  cents  per  lb. ;  250  lbs.  of  cheese, 
at  8  cents  per  lb. ;  154  lbs.  of  butter,  at  15  cents  per  lb. : 


V  119.       ^  MISCELLANEOUS    EXAMPLES.  239 

in  pay  he  received  60  lbs.  of  sugar,  at  10  cents  per  lb. ;  15 
gallons  of  molasses,  at  42  cents  per  gallon ;  ^  barrel  of  mack- 
erel, at  $  3'75  J  4  bushels  of  salt,  at  $  1'25  per  bushel ;  and 
the  balance  in  money :  how  much  money  did  he  receive  ? 

Ans.    $68'85. 
22.  A  farmer  carried  his  grain  to  market,  and  sold 
75  bushels  of  wheat,  at  $  1'45  per  bushel. 

64 rye,      ...  $   *95 

142 com,    ...  $    '50 

In  exchange  he  received  sundry  articles  : — 

8  pieces  of  cloth,  each 

containing  31  yds.,  at  $  V75  per  yd. 

2  quintals  of  fish,      ...  $  2^30  per  quin. 

8hhds.    of  salt,  ...  $4^30  per  hhd. 


and  the  balance  in  money. 

How  much  money  did  he  receive  ?  Ans,    $38*80. 

23.  A  man  exchanges  760  gallons  of  molasses,  at  37j^ 
cents  per  gallon,  for  66^  cwt.  of  cheese,  at  $  4  per  cwt. ; 
how  much  will  be  the  balance  in  his  favour?        ^t?^.    $  19. 

24.  Bought  84  yards  of  cloth,  at  $V25  perya^d;  ho\% 
much  did  it  come  to  ?  How  many  bushels  of  wheat,  a 
$  VoQ  per  bushel,  will  it  take  to  pay  for  it? 

Am.  to  the  lasty  70  bushels- 

25.  A  man  sold  342  pounds  of  beef,  at  6  cents  per  pound, 
and  received  his  pay  in  molasses,  at  37J  cents  per  gallon ; 
how  many  gallons  did  he  receive  ?  Arts,  54'72  gallons. 

26.  A  man  exchanged  70  bushels  of  rye,  at  $  '92  per 
bushel,  for  40  bushels  of  wheat,  at  $  1^37 J-  per  bushel,  and 
received  the  balance  in  oats,  at  $  '40  per  bushel ;  how 
many  bushels  of  oats  did  he  receive  ?  Ans.  23^. 

27.  How  many  bushels  of  potatoes,  at  1  s.  6  d.  per  bushel, 
must  be  given  for  32  bushels  of  barley,  at  2  s.  6  d.  per 
bushel  ?  Ans.  63^  bushels. 

28.  How  much  salt,  at  $  1'50  per  bushel,  must  be  given 
in  exchange  for  15  bushels  of  oats,  at  2  s.  3  d.  per  bushel  ? 

Note.  It  will  be  recollected  that,  when  the  price  and  cost 
lire  given,  to  find  the  quantity-,  they  must  both  be  reduced  to 
the  same  denomination  before  dividing.        Ajls.  3  J  bushels. 

29.  How  much' wine,  at  $2^75  per  gallon,  must  be  givon 
in  exchange  for  40  yards  ©f  cloth,  at  7  ».  6  d.  per  y^d  ? 

Ans.  18^  gallon*. 


MO  MISCELLA?fEOUS    EXAMPLES.  IT  119. 

30.  A  had  41  cwt.  of  hops,  at  30  s.  per  cvvt,  for  which 
B  gave  him  20  £ .  in  money,  and  the  rest  iu  prunes,  at  5  d. 
per  lb. ;  how  many  prunes  did  A  receive  ? 

Ans.  17  cwt.  3  qrs.  4  lbs. 

81.  A  has  linen  cloth  worth  $'30  per  yard;  but,  in  bar- 
tering, he  will  have  $  '35  per  yard  ;  B  has  broadcloth  worth 
$  3"75  ready  money ;  at  what  price  ought  the  broadcloth 
to  be  rated  in  bartering  with  A  ? 

'30  :  '35  :  :  3'75  :  $  4'375,  Aiis,  Or,  i|^  of  3'75  = 
$  4'37^,  Ans,  The  two  operations  will  be  seen  to  be  ex- 
actly alike. 

32.  If  cloth,  worth  2  s.  per  yard,  cash,  be  rated  in  barter 
at  2  s.  6  d.,  how  should  wheat,  worth  8  s.  cash,  be  rated  in 
exchanging  for  the  cloth  ?  Ans,  10  s.,  or  $  1'666§. 

33.  If  4  bushels  of  corn  cost  $  2,  what  is  it  per  bushel  ? 

Ans.    $'50. 

34.  If  9  bushels  of  wheat  cost  $  13'50,  what  is  that  per 
bushel?  Ans,   $  1'50. 

35.  If  40  sheep  cost  $  100,  what  is  that  per  head  ? 

Ans,    $2'50. 

36.  If  3  bushels  of  oats  cost  7  s.  6  d.,  how  much  are  they 
per  bushel  ?  Ans,  2  s.  6  d.,  =  $  '41f . 

37.  If  22  yards  of  broadcloth  cost  21  £ .  9  s.,  what  is  the 
price  per  yard  r  An^,  19  s.  6  d.,  =  $  3'25; 

38.  At  $  '50  per  bushel,  how  much  corn  can  be  bought 
for  $  2'00  ?  Ans,  4  bushels. 

39.  A  man,  having  $  100,  would  lay  it  out  in  sheep,  at 
$  2'50  apiece  ;  how  many  can  he  buy  ?  Ans,  40. 

40.  If  20  cows  cost  $  300,  what  is  the  price  of  1  cow  ? 
of  2  cows  ?  of  5  cows  ?  of  15  cows  ? 

Ans.  to  the  last^  $  225. 

41.  If  7  men  consume  24  lbs.  of  meat  in  one  week,  how 

much  would  1  man  consume  in  the  same  time  ? 2  men  ? 

5  men  ?   10  men  ?  Ans,  to  the  lasty  34^  lbs. 

Note,  Let  the  pupil  also  perform  these  questions  by  the 
rule  of  proportion. 

42.  If  I  pay  $  6  for  the  use  of  $  100,  how  much  must  I 
pay  for  the  use  of  $  75  ?  Ans,    $  4'50. 

4?.  What  premium  must  I  pay  for  the  insurance  of  my 
house  against  loss  by  fire,  at  the  rate  of  ^  per  cent.,  that  is, 
^  dollar  on  a  hundred  dollars,  if  my  house  be  valued  at 
$2475?  ^715.    $12'3W. 


IT  119.  MISCELLANEOUS    EXAMPLES,  241 

44.  What  will  be  the  insurance,  per  annum,  of  a  store  and 
contents,  valued  at  $9S76'40,  at  1^  per  centum? 

Am,    $148446. 

45.  What  commission  must  I  receive  for  selling  $478 
worth  of  books,  at  8  per  cent  ?  Ans.    $  38^24. 

46.  A  merchant  bought  a  quantity  of  goods  for  $734, 
and  sold  them  so  as  to  gain  21  per  cent. ;  how  much  did  he 
gain  ?  and  for  how  much  did  he  sell  his  goods  ? 

Ans,  to  the  last^  $  88844. 

47.  A  merchant  bought  a  quantity  of  goods  at  Boston,  for 
$  500,  and  paid  $  43  for  their  transportation ;  he  sold  them 
so  as  to  gain  24  per  cent,  on  the  whole  cost ;  for  how  much 
did  he  sell  them  ?  Ans.    $  673^32. 

48.  Bought  a  quantity  of  books  for  $  64,  but  ior  cash  a 
discount  of  12  per  cent,  was  made ;  what  did  the  books 
cost?  Ans,    $56^32. 

49.  Bought  a  book,  the  price  of  which  was  marked 
$  4'50,  but  for  cash  the  bookseller  will  sell  it  at  33^  per 
cent,  discount ;  what  is  the  cash  price  ?  Ans.    $  3^00. 

50.  A  merchant  bought  a  cask  of  molasses,  containing  120 
gallons,  for  $  42 ;  for  how  much  must  he  sell  it  to  gain  15 
per  cent.  ?  how  much  per  gallon  ?  Aus.  to  last,  $  '40^. 

51.  A  merchant  bought  a  cask  of  sugar,  containing  740 
pounds,  for  $  59^20 ;  how  must  he  sell  it  per  pound,  to  gain 
25  per  cent.  ?  Ans.    $  40. 

52.  What  is  the  interest,  at  6  per  cent,  of  $  71^02  for  17 
months  12  days  ?  Ans,    $  6478  -|-. 

53.  What  is  the  interest  of  $487*003  for  18  months  ? 

Ans.    $43'83+. 

54.  What  is  the  interest  of  $  8*50  for  7  months  ? 

Ans.    $<297J. 
65.  What  is  the  interest  of  $  1000  for  5  days  ? 

Ans.    $*833f 
5d.  What  is  the  interest  of  $  *50  for  10  years  ? 

Ans.   $*30. 

57.  What  is  the  interest  of  $84*25  for  15  montlis  and  7 
days,  at  7  per  eent.  ?  Ans.    $  7*486  -f-- 

58.  What  is  the  interest  of  $  154*01  for  2  years,  4  months 
and  3  days,  at  5  piT  cent.  ?  Am.    $  18*032. 

59.  What  sum,  put  to  interest  at  6  per  cent.,  will,  in  2 
years  and  6  months,  amount  to  $  150  ? 

Note.     See  IT  85.  Am.    $  130*134  -V-. 

60.  I  owe  a  man  $  475*50,  to  be  paid  in  16  months  witW- 

X 


242  MISCELLANEOUS    EXAMPLES.  IT  11 9, 

out  interest;  what  is  the  present  worth  of  that  debt,  the  use 
of  the  money  being  worth  6  per  cent.  ?     Ans,    $  440'277  -|- 

61.  What  is  the  present  worth  of  $  1000  payable  in  4 
years  and  2  months,  discounting  at  the  rate  of  6  per  cent.  ? 

Ans,    $800. 

62.  A  merchant  bought  articles  to  the  amount  of  $  500, 
and  sold  them  for  $  575  ;  how  much  did  be  gain  ? 

What  per  cent,  was  his  gain  ?  that  is,  How  many  dollars 
did  he  gain  on  each  $100  which  he  laid  out?  If  $500 
gain  $75,  what  dof^  $  100  gain?  Am.   15  percent. 

63.  A  merchant  bought  cloth  at  $  3^50  per  yard,  and  sold 
it  at  $  4'25  per  yard ;  how  much  did  he  gain  per  centum  ? 

Ans,  21^  per  cent. 

64.  A  man  bought  a  cask  of  wine,  containing  126  gallons, 
for  $  283'50,  and  sold  it  out  at  the  rate  of  $  2^75  per  gal- 
lon ;  how  much  was  his  whole  gain  ?  how  much  per  gal- 
lon ?     how  much  per  cent.  ? 

Am,  His  whole  gain,  $63'00;  per  gallon,  $'50;  which 
is  22f  per  centum. 

65.  If  $  100  gain  $  0  in  12  months,  in  what  time  will  it 
gain  $  4  ?  $  10  ?  $  14  ? 

Ans,  to  the  last^  28  months. 

66.  In  what  time  will  $  54'50,  at  6  per  cent.,  gain  $  248  ? 

Ans.  8  months. 

67.  20  men  built  a  certain  bridge  in  60  days,  but,  it  being 
carried  away  in  a  freshet,  it  is  required  how  many  men  can 
rebuild  it  in  50  days. 

(lays.        days.  men. 

50    :   60    :  :   20    :   24  men,  Am. 

68.  If  a  field  will  feed  7  horses  8  weeks,  how  long  will  it* 
feed  28  horses  ?  Am,  2  weeks. 

69.  If  a  field,  20  rods  in  length,  must  be  8  rods  in  width 
to  contain  an  acre,  how  much  in  width  must  be  a  field,  16 
rods  in  length,  to  contain  the  same?  Ans,   10  rods. 

70.  If  I  purchase  for  a  cloak  12  yards  of  plaid  |-  of  a  yard 
wide,  how  much  hocking  l^  yards  wide  must  I  have  to  line  it  ? 

Am,  5  yards. 

71.  If  a  man  earn  $75  in  5  months,  how  long  must  he 
work  to  earn  $  460  ?  Ans.  30|  months. 

*72.  A  owes  B  $  540,  but,  A  not  being  worth  so  much 
money,  B  agrees  to  take  $  '75  on  a  dollar  ;  what  sinn  must 
B  receive  for  the  debt  ?  Ans,    $  406. 

73.  A  cistern,  whose  capacity  is  400  galloas,  is  supplied 


V  119.  MISCELLANEOUS    EXAMPLES.  243 

by  a  pipe  which  lets  in  7  gallons  in  5  minutes ;  but  there  is 
a  leak  in  the  bottom  of  the  cistern  which  lets  out  2  gallons 
in  6  minutes ;  supposing  the  cistern  empty,  in  what  time 
would  it  be  filled  r 

In  1  minute  ^  of  a  gallon  is  admitted,  but  in  the  same  time 
f  of  a  gallon  leaks  out.  Ans,  6  hours,  15  minutes. 

74.  A  ship  has  a  leak  which  will  fill  it  so  as  to  make  it 
sink  in  10  hours ;  it  has  also  a  pump  which  will  clear  it  in 
15  hours  :  no^v,  if  they  begin  to  pump  when  it  begins  to 
leak,  in  what  time  will  it  sink  ? 

In  1  hour  tl>e  ship  would  be  -£^  filled  by  the  leak,  but  in 
die  same  time  it  would  be  -^  emptied  by  the  pump. 

Alls,  30  hours. 

75.  A  cistern  is  supplied  by  a  pipe  which  will  fill  it  in 
40  minutes ;  how  many  pipes,  of  the  same  bigness,  will  fill  it 
in  5  minutes  ?  Ans,  8. 

76.  Suppose  I  lend  a  friend  $500  for  4  months,  he 
promising  to  do  me  a  like  favour;  some  time  afterward,  I 
have  need  of  $  300 ;  how  long  may  I  keep  it  to  balance  the 
former  favour  ?  Ans,  6f  months. 

77.  Suppose  800  r5oldiers  were  in  a  garrison  with  pro- 
visions sufficient  for  2  months ;  how  many  soldiers  must  de- 
part, that  the  provisions  may  serve  them  5  months  ? 

Ans,  480. 

78.  If  my  horse  and  saddle  are  worth  $  84,  and  my  horse 
be  worth  6  times  as  much  as  my  saddle,  pray  what  is  the 
value  of  my  horse  ?  Ans,    $  72. 

79.  Bought  45  barrels  of  beef,  at  $  3^50  per  barrel,  among 
which  are  16  barrels,  whereof  4  are  worth  no  more  than  3 
of  the  others;  how  much  must  t  pay  ?  Ans,    $  143*50. 

80.  Bought  126  gallons  of  rum  for  $110;  how  much 
v;ater  must  be  added  to  reduce  the  first  cost  to  $  '75  per 
gallon  ? 

Note,  If  $  '75  buy  1  gallon,  how  many  gallons  will  $  110 
buy?  Ans,  20§  gallons. 

81.  A  thief,  bavins:  24  miles  start  of  the  officer,  holds  his 
way  at  the  rate  of  6  miles  an  h^ur ;  the  officer  pressing  on 
after  him  at  the  rate  of  8  miles  an  hour,  how  much  does  he 
gain  in  1  hour  ?  how  long  before  he  will  overtake  the  thief? 

Ans,   12  hours. 

82.  A  hare  starts  12  rods  before  a  hound,  but  is  not  per- 
ceived by  him  till  she  has  been  up  H  minutes;  she  scuds 
away  at  the  rate  of  36  rods  a  minute,  and  the  doir,  on  view. 


244  MISCELLANEOUS    EXAMPLES.  IT   iW^ 

makes  after,  at  the  rate  of  40  rods  a  rriinutc ;  how  long  will 
the  course  hold  ?  and  what  distance  will  the  dog  run  ? 

Am,  14^  minutes,  and  he  will  run  570  roda. 

83.  The  hour  and  minute  hands  of  a  watch  are  exactly 
together  at  12  o'clock ;  when  are  they  next  together  ? 

In  1  hour  the  minute  hand  passes  over  12  spaces,  and  the 
hour  hand  over  1  space  ;  that  is,  the  minute  hand  gains  upon 
the  hour  hand  1 1  spaces  in  1  hour ;  and  it  must  gain  12 
spaces  to  coincide  with  it.  Ana.   1  h.  6  m.  2T^  s. 

84.  There  is  an  island  20  miles  in  circumference,  and 
three  men  start  together  to  travel  the  same  way  about  it ;  A 
goes  2  miles  per  hour,  B  4  miles  per  hour,  and  C  6  miles 
per  hour  j  in  what  time  will  they  come  together  again  ? 

Ans.  10  hours. 

85.  There  is  an  island  20  miles  in  circumference,  and 
two  men  start  together  to  travel  around  it;  A  travels  2  miles 
per  hour,  and  B  6  miles  per  hour  ;  how  long  before  they  will 
again  come  together  ? 

B  gains  4  miles  per  hour,  and  must  gain  20  miles  to  over- 
take A  ;  A  and  B  will  therefore  be  together  once  in  eveiy 
5  hours. 

86.  In  a  river,  supposing  two  boats  start  at  the  same  time 
from  places  300  miles  apart ;  the  one  proceeding  up  stream 
is  retarded  by  the  current  2  miles  per  hour,  while  that  mov- 
ing down  stream  is  accelerated  the  same ;  if  both  be  pro- 
pelled by  a  steam  engine,  which  would  move  them  8  miles 
per  hour  in  still  water,  how  far  from  each  starting  place  will 
the  boats  meet  ? 

Ans.  112|  miles  from  the  lower  place,  and  1 87 j  miles 
from  the  upper  place, 

87.  A  man  bought  a  pipe  (126  gallons)  of  wine  for 
$275;  he  Vv^ishes  to  fill  10  bottles,  4  of  which  contain  2 
quarts,  and  6  of  them  3  pints  each,  and  to  sell  the  remainder 
so  as  to  make  30  per  cent,  on  the  first  cost ;  at  what  rate 
per  gallon  must  he  sell  it  ?  Ans.    $  2^936  -f-- 

88.  Thomas  sold  150  pine  apples  at  $'33^  apiece,  and 
received  as  much  i.iCney  as  Harry  received  for  a  certain 
number  of  watermelons  at  $  '25  apiece ;  how  much  money 
did  each  receive,  and  how  many  melons  had  Harry  ? 

Ans.    $  50,  and  200  melons. 

89.  The  third  part  of  an  army  was  killed,  the  fourth  pari 
taken  prisoners,  and  1000  fled ;  how  many  were  in  this  army? 

This  and  the  eighteen  following  questions  are  usually 


V   119.  MISCELLANEOirS    EXAMPLES.  245 

wrought  by  a  rule  called  Position,  but  they  are  more  easily 
solved  on  general  principles.  Thus,  ^  -j-  i  ^=^  t^j  ^^  ^^ 
army;  therefore,  1000  is  ^^^  ^^  the  whole  number  of  men; 
and,  if  5  twelfths  be  1000,  how  much  is  12  twelfths,  or  the 
whole  ?  -A^is.  2400  men. 

90.  A  farmer,  being  asked  how  many  sheep  he  had,  an- 
swered, that  he  had  them  in  5  fields ;  in  the  first  were  ^  of  his 
flock,  in  the  second  J,  in  the  third  |,  in  the  fourth  y^,  and 
in  the  fifth  450  ;  how  many  had  he  ?  Ans.   1200. 

91.  There  is  a  pole,  ^  of  which  stands  in  the  mud,  ^  in 
the  water,  and  the  rest  of  it  out  of  the  water;  required  the 
part  out  of  the  water.  Ans.  y^. 

92.  If  a  pole  be  -J  in  the  mud,  |-  in  the  water,  and  6  feet 
out  of  the  water,  what  is  the  length  of  the  pole  ?  Ans.  90  feet. 

93.  The  amount  of  a  certain  school  is  as  follows :  ^^  of 
the  pupils  study  grammar,  f  geography,  y^j  arithmetic,^ 
learn  to  write,  and  9  learn  to  read  :  what  is  the  number  of 
each  ? 

Ans,  5  in  grammar,  30  in  geography,  24  in  arithmetic ; 
12  learn  to  write,  and  9  learn  to  read. 

94.  A  man,  driving  his  geese  to  market,  was  met  by 
another,  who  said,  "  Good  morrow,  sir,  with  your  hundred 
geese  ;"  says  he,  '^  I  have  not  a  hundred;  but  if  I  had,  in  ad' 
dition  to  my  present  number,  one  half  as  many  as  I  now 
have,  and  2^  geese  more,  I  should  have  a  hundred :"  how 
many  had  he  ? 

100  —  2^  is  what  part  of  his  present  number  ? 

Am.  He  had  65  geese. 

95.  In  an  orchard  of  fruit  trees,  ^-  of  them  bear  apples, 
^  pears,  ^  plums,  60  of  them  peaches,  and  40,  cherries ; 
how  many  trees  does  the  orchard  contain  ?  Ans,   1200. 

96.  In  a  certain  village,  ^  of  the  houses  are  painted  white, 
J-  red,  ^  yellow,  3  are  painted  green,  and  7  are  unpainted; 
how  many  houses  in  the  village  ?  Ans,   120. 

97.  Seven  eighths  of  a  certain  number  exceed  four  fifths  of 
the  same  number  by  6  ;  required  the  number. 

^  —  ^=z  ^\ ;  consequently,  6  is  ^ry  of  the  required  num- 
ber. Ans,  80. 

98.  What  number  is  that^  to  which  if  ^  of  itself  be  added, 
(he  sum  will  be  30  ?  Ans,  2d. 

99.  What  number  is  that,  to  which  if  its^  and  I  be  added, 
the  sum  will  be  84  ^ 

84=l+^-f-^zz:J  times  the  required  number.   Atis,  48. 
X* 


246  MISCELLANEOUS   EXAMPLES.  If   11^ 

100.  What  number  is  that,  which,  being  increased  by  f  and 
I  of  itself,  and  by  22  more,  will  be  made  three  times  &s 
much  ? 

The  number,  being  taken  1,  f,  and  f  times,  will  make  2^ 
times,  and  22  is  evidently  what  that  wants  of  3  times. 

Am,  30. 

101.  What  number  is  that,  which  being  increased  by  f,  | 
and  |-  of  itself,  the  sum  will  be  234f  ?  Ans.  90. 

102.  A,  B,  and  C,  talking  of  their  ages,  B  said  his  age 
was  once  and  a  half  the  age  of  A,  and  C  said  his  age  was 
twice  and  one  tenth  the  age  of  both,  and  that  the  sum  of 
their  ages  was  93  ;  what  was  the  age  of  each  ? 

Ans,  A  12  years,  B  18  years,  C  63  years  old. 

103.  A  schoolmaster,  being  asked  how  many  scholars  he 
had,  said,  "  U  I  had  as  many  more  as  I  now  have,  J  as  many, 
^  as  many,  -^  and  {  as  many,  I  should  then  have  435  ;"  what 
was  the  number  of  his  pupils  ?  Ans,  120l 

104.  A  and  B  commenced  trade  v/ith  equal  sums  of 
money ;  A  gained  a  sum  equal  to  -^  of  his  stock,  and  B  lost 
$  200 ;  then  A's  money  was  double  that  of  B's ;  what  was 
the  stock  of  each  ? 

By  the  condition  of  the  question,  one  half  of  f,  that  is,  ^ 
of  the  stock,  is  equal  to  -I  of  the  stock,  less  $  200  ;  conse- 
quently, $  200  is  f  of  the  stock.  Ans,  $  500. 

105.  A  man  was  hired  50  days  on  these  conditions, — that, 
for  every  day  he  worked,  he  should  receive  $  '75,  and,  for 
every  day  he  was  idle,  he  should  forfeit  $  '25 ;  at  the  ex- 
piration of  the  time,  he  received  $  27'50  ;  how  many  days 
did  he  work,  and  how  many  was  he  idle  ? 

Had  he  worked  every  day,  his  wages  would  have  been 
$  '75  X  50  131  $  37'50,  that  is,  $  10  more  tlian  he  received ; 
but  everyday  he  wa.s  idle  lessened  his  wages  $  '75  -j-  $  ''^^ 
=:=  $  1  ;  consequently  he  was  idle  10  days. 

Ans,  He  wrought  40,  and  was  idle  10  days. 

106.  A  and  B  have  the  same  income;  A  saves  -J  of  his; 
but  B,  by  spending  $  30  per  annum  more  than  A,  at  the  end 
of  8  years  finds  himself  $  40  in  debt ;  what  is  their  income, 
and  what  does  each  spend  per  annum  ? 

Ans,  Their  income,  $200pjr  annum;  A  spends  $175, 
and  B  $205  per  unniim. 

107.  A  man,  Ijing  at  tlic  p^nnt  of  death,  left  to  his  three 
ious  his  property ;  to  A  J  wanting  $  20,  to  B  ^,  and  to  C 


IT  119.  MISCELLANEOUS    EXAMPLES.  2i1f 

the  rest,  which  was  $  10  less  than  the  share  of  A ;   what 
was  each  one's  share  ?  Ans.  $  SO,  $  50.and  $  70. 

108.  There  is  a  fish,  whose  head  is  4  feet  long; ;  his  tail 
is  as  long  as  his  head  and  ^  the  length  of  his  body,  and  his 
body  is  as  long  as  his  head  and  tail;  what  is  tlie  length  of 
the  fish  ? 

The  pupil  will  perceive,  that  the  length  of  the  body  is  ^ 
the  length  of  the  fish.  Ans.  32  feet 

109.  A  can  do  a  certain  piece  of  work  in  4  days,  and  B 
can  do  the  same  work  in  3  days ;  in  what  time  would  both, 
working  togethei,  perform  it  ?  Ans.  1^  days. 

110.  Three  persons  can  perform  a  certain  piece  of  work  in 
the  following  manner  :  A  and  B  can  do  it  in  4  days,  B  and 
C  in  6  days,  and  A  and  C  in  5  days  :  in  what  time  can  they 
all  do  it  together  ?  A7is.  3-^^  days. 

111.  A  and  B  can  do  a  piece  of  work  in  5  days ;  A  can  do 
it  in  7  days  ;   in  how  many  days  can  B  do  it  ?     Aiis.  17J^  days. 

112.  A  man  died,  leaving  $  1000  to  be  divided  between 
his  two  sons,  one  14,  and  the  other  18  years  of  age,  in  such 
proportion,  that  the  share  of  each,  being  put  to  interest  at  6 
per  cent.,  should  amount  to  the  same  sum  when  they  should 
arrive  at  the  age  of  21 ;   wliat  did  each  receive  ? 

^W5.  The  elder,  $546453+ ;  the  younger,  $453^846+. 

113.  A  house  being  let  upon  a  lease  of  5  years,  at  $60 
per  annum,  and  the  rent  being  in  arrear  for  the  whole  time, 
what  is  the  sum  due  at  the  end  of  the  term,  simple  interest 
being  allowed  at  6  per  cent.  ?  A7is.  $  336. 

114.  If  3  dozen  pair  of  gloves  be  equal  in  value  to  40 
yards  of  calico,  and  100  yards  of  calico  to  three  pieces  of 
satinet  of  30  yards  each,  and  the  satinet  be  worth  50 
cents  per  yard,  how  many  pair  of  gloves  can  be  bought  for 
$  4  ?  Ans.  8  pair. 

115.  A,  B,  and  C,  would  divide  $  100  between  them,  so 
as  that  B  may  have  $3  more  than  A,  and  C$4  more  than 
B  ;    how  much  must  each  man  have  ? 

Ans.  A  $  30,  B  $  33,  and   C  $  37. 

116.  A  man  has  pint  bottles,  and  half  pint  bottles;    how 

much  wine  will  it  take  to  fill  1  of  each  sort  ?  how 

much  to  fill  2  of  each  sort  ? how  much  to  fill  6  of  each 

sort? 

117.  A  man  would  draw  off  30  gallons  of  wine  into  1 
nint  and  2  pint  bottles,  of  each  an   equal  number;   how 


248  MISCELLANEOUS   EXAMPLES.  IT   119. 

many  bottles  will  it  take,  of  each  kind,  to  contaia  the  30 
gallons  ?  ^  Ans,  80  of  each, 

118.  A  merchant  has  canisters,  some  holdings  pounds, 
some  7  pounds,  and  some  12  pounds  ;  how  many,  of  each 
an  equal  number,  can  be  filled  out  of  12  cwt.  3  qrs.  12  lbs. 
of  tea  ?  Ans.  60. 

119.  If  IS  grains  of  silver  make  a  thimble,  and  12  pwts. 
make  a  teaspoon,  how  many,  of  each  an  equal  number,  can 
be  made  fromlSoz.  6  pwts.  of  siher?  Ans.  24  of  each. 

120.  Let  60  cents  be  divided  among  three  boys,  in  such 
a  manner  that,  as  often  as  the  first  has  3  c^^nts,  the  second 
shall  have  5  cents,  and  the  third  7  cents ;  how  many  cents 
will  each  receive  ?  Ans.  12,  20,  and  28  cents. 

121.  A  gentleman,  having  50  shillings  to  pay  among  his 
labourers  for  a  day's  work,  would  give  to  every  boy  6  d.,  to 
every  woman  8  d.,  and  to  every  man  16  d. ;  the  number  of 
boys,  women,  and  men,  was  the  same  ;  I  demand  the  number 
of  each.  Ans.  20. 

122.  A  gentleman  had  7  £.  17  s.  6  d.  to  pay  among  his 
labourers ;  to  every  boy  he  gave  6  d.,  to  every  woman  8  d., 
and  to  every  man  16  d. ;  and  there  were  for  every  boy  three 
women,  and  for  every  woman  two  men ;  I  demand  the  num- 
ber of  each.  Ans.   15  boys,  45  women,  and  90  men. 

123.  A  farmer  bought  a  sheep,  a  cow,  and  a  yoke  of  oxen 
for  $  82'50  ;  he  gave  for  the  cow  8  times  as  much  as  for 
the  sheep,  and  for  the  oxen  3  times  as  much  as  for  the  cow ; 
how  much  did  he  give  for  each  ? 

Ans.  For  the  sheep  $  2'50,  the  cow  $  20,  and  the  oxen 
$60. 

124.  There  was  a  farm,  of  which  A  owned  f,  and  B  ^, 
the  farm  was  sold  for  $  1764;  what  was  each  one's  share 
of  the  money  ?  Ans.  A's  $  504,  and  B's  $  1260. 

125.  Four  men  traded  together  on  a  capital  of  $3000,  of 
w^hich  A  put  in  ^,  B  ^,  C  ^,  and  D  y^  ;  at  the  end  of  3  years 
they  had  gained  $2364;  what  was  each  one's  share  of  the 
gain?  (  A's  $1182. 

.        )  B's  $    591. 

^"^*   J  C's  $    394. 

(  D's  $    197. 

126.  Three  merchants  accompanied  ;  A  furnished  f  of 
the  capital,  B  f,  and  C  the  rest;  tliey  gain  $1250;  what 


i[  119.  MISCELLANEOUS   EXAMPLES.  249 

part  of  the  capital  did  C  furnish,  and  what  is  each  One'rf 
share  of  the  gain  ? 

Ans.  C  furnished  -/j^  of  the  capital ;  and  A's  share  of  the 
gam  was  $500,  B's  $  468'75,  and  C's  $281'25. 

127.  A,  B,  and  C,  traded  in  company;  A  put  in  $  500,  B 
$f350,  and  C  120  yards  of  cloth  ;  they  gained  $332^50,  of 
which  C's  share  was  $  120  ;  what  was  the  value  of  C's 
cloth  per  yard,  and  what  was  A  and  B's  shares  of  the 
gain  ? 

Note,  C's  gain,  heing  $  120,  is  f  JJg^-  =  y^j  of  the  whole 
gain :  hence  the  gain  of  A  and  B  is  readily  found ;  also  the 
price  at  w^hicli  C's  cloth  was  valued  per  yard. 

(C's  cloth,  per  yard,  $4. 
Am,  <  A's  share  of  the  gain,  $  125. 
f  B's  do.    g  87^50. 

128.  Three  gardeners,  A,  B,  and  C,  having  bought  a 
piece  of  ground  find  the  profits  of  it  amount  to  i20iS.  per 
annum.  Now  the  sum  of  money  which  they  laid  down  was 
in  such  proportion,  that,  as  often  as  A  paid  5£ .,  B  paid  7£ ., 
and  as  often  as  B  paid  4ciC.,  C  paid  6JS.  I  demand  hovt^ 
much  each  man  must  have  per  aniium  of  the  gain. 

Note.  By  the  question,  so  often  a-^  A  paid  5  j£ .,  C  paid  f  of 
7£.       Ans,  A2G£,  13  s.  4  d.,  B  37je.  6  s.  8  d.,  C  b6£. 

129.  A  gentleman  divided  his  fortune  among  his  so~#Sy 
giving  A  9i£.  as  often  as  B  b£,,  and  C  3j2.  as  often  as  B 
7£,j  C's  dividend  was  1537|J^.;  to  what  did  the  whole 
estate  amount?  ^1//^.   11583ciK.  S  s.  10  d. 

130.  A  and  B  undertake  a  piece  of  work  for  ^^54,  on 
which  A  employed  3  hands  5  days,  and  I^  employed  7  hands 
3  days ;  what  part  of  the  work  was  done  by  A,  Vv^hat  part 
by  B,  and  what  was  each  one's  share  of  the  money  ? 

Ans.  A-j^,  and  B-/2-;  A's  money  $22^50,  B's    $3V50. 

131.  A  and  B  trade  in  company  for  one  year  only;  on 
the  tirst  of  January,  A  put  in  $  1200,  but  B  could  not 
put  any  money  into  the  stock  until  the  first  of  April;  what 
did  he  then  put  in,  to  have  an  equal  share  with  A  at  the  end 
of  the  year?  Ans,    $  1600. 

132.  A,  B,  C,  and  D,  spent  35  s.  at  a  reckoning,  and,  be- 
ing a  little  dipped,  agreed  that  A  should  pay  f ,  B  ^,  C  ^, 
and  D  ^ ;  wliat  did  each  pav  in  this  proportion  ? 

Ans,  A  13  s.  4  d.,  B  10  s.,   C  6  s.  8  d.,  and  D  5  s. 

133.  There  are  3  horses,  belonging  to  3  men,  employed  to 
draw  a  load  of  plaster  from  Boston  to  Windsor  for  $26'45; 


250  MISCELLANEOUS    EXAMPLES.  IT   119/ 

A  and  B's  horses  together  are  supposed  to  do  |  of  th« 
work,  A  and  C's  j^^j,  B  and  C's  ^g  ;  they  are  to  be  paid 
proportionally ;  what  is  each  one's  share  of  the  money  ? 

(  A's  $ll'oO  (=^^0 
Ans,  \  B's  $    5^75  {=z  J^,) 
(  C's  $    9^20  {=  ^V) 
Proof, $  26'45. 

134.  A  person,  who  was  possessed  of  f  of  a  vessel,  sold 
1^  of  his  share  for  375  iS . ;  what  was  the  vessel  worth  ? 

Aiis,  1500£. 

135.  A  gay  fellow  soon  got  the  better  of  f  of  his  for- 
tune ;  he  then  gave  1500^ .  for  a  commission,  and  his  profu- 
sion continued  till  he  had  but  450c£.  left,  which  he  found  to 
be  ju"t  f  of  his  money,  after  he  had  purchased  his  commis- 
sion ;  what  was  his  fortune  at  lirsl?  Ans,  3780^. 

136.  A  younger  brother  received  1560iB .,  which  was  just  y^ 
of  his  elder  brother's  fortune,  and  of  times  the  elder  brother's 
fortune  was  |  as  much  again  as  the  father  was  ^vorth  ;  pray, 
what  was  the  value  of  his  estate  ?    Ans.  19165  jB .  14  s.  3^^  d. 

137.  A  gentleman  left  his  son  a  fortune,  y^^  of  which  he 
spent  in  three  months;  f  of  -|  of  the  remainder  lasted  him 
nine  months  longer,  when  he  had  only  537 £.  left;  what 
was  the  sum  bequeathed  him  by  his  father  ? 

Ans.  2082  i3.  18  s.  2-j\  d. 

138.  A  cannon  ball,  at  the  first  discharge,  flies  about  a 
mile  in  eujht  seconds ;  at  this  rate,  how  long  would  a  ball 
be  in  p.^ssing  from  the  earth  to  the  sun,  it  being  95173000 
miles  distant? 

Alls.  24  years,  46  days,  7  hours,  33  minutes,  20  seconds. 

139.  A  general,  disposing  his  army  into  a  square  battalion, 
found  he  had  231  over  and  above,  but,  increasing  each  side 
with  one  soldier,  he  wanted  44  to  fill  up  the  square  ;  of  how 
many  men  did  his  army  consist?  Am.   19000. 

140.  A  and  B  cleared,  by  an  adventure  at  sea,  45  guineas, 
which  was  35 iB.  per  cent,  upon  the  money  advanced,  and 
with  which  they  agreed  to  purchase  a  genteel  horse  and 
carriage,  whereof  they  were  to  have  the  use  in  proportion 
to  the  sums  adventured,  which  was  found  to  be  11  to  A,  as 
often  as  8  to  B ;  what  money  did  each  adventure  ? 

Ans.  A  104£.  4  s.  2-\^-  d.,  B  75£.  15  s.  9-9^  d. 

141.  Tubes  may  be  made  of  gold,  weighing  not  more 
than  at  the  rate  of  j-^^-^  of  a  grain  per  foot  ?  what  would  be 
the  weight  of  such  a  tube,  which  would  extend  across  the 


IT   119.  MISCELLANEOUS    EXAMPLES.  S51 

Atlantic  from  Boston  to  London,  estimating  the  distance  a| 
3000  miles  ?  Aiis.  1  lb.  8  oz.  6  pwts.  3^^  gra 

142.  A  military  officer  drew  up  his  soldiers  in  rank  and 
file,  having  the  number  in  rank  and  file  equal ;  on  being 
reinforced  with  three  times  his  first  number  of  men,  he 
placed  them  all  in  the  same  form,  and  then  the  number  in 
rank  and  file  was  just  double  what  it  was  at  first ;  he  was 
again  reinforced  with  three  times  his  whole  number  of  men, 
and,  after  placing  tbem  all  in  the  same  form  as  at  first,  his 
number  in  rank  and  file  was  40  men  each ;  hi>vv  many  men 
had  he  at  first  ?  Ans,  100  men. 

143.  Supposing  a  man  to  stand  SO  feet  from  a  steepje,  and 
that  a  line  reaching  from  the  belfry  to  the  man  is  just  100 
feet  in  length ;  the  top  of  the  spire  is  3  times  as  high  above 
the  ground  as  the  steeple  isj  what  is  the  height  of  the 
spire  ?  and  the  length  of  a  line  reaching  from  the  top  of  th« 
spire  to  the  man?     See  IT  109. 

Ans.  to  last  J  197  feet,  nearly. 

144.  Two  ships  sail  from  the  same  port ;  one  sails  direct- 
ly east,  at  the  rate  of  10  miles  an  hour,  and  the  other  direct- 
ly south,  at  the  rate  of  7^  miles  an  hour;  how  many  miles 

apart  will  they  be  at  the  end  of  1   hour  ? 2  hours  ? 

24  hours  ? 3  days  ?  Ans.  to  last^  900  miles. 

145.  There  is  a  square  field,  each  side  of  which  is  60 
rods ;  what  is  the  distance  between  opposite  corners  ? 

Ans.  70*71  +  rods. 

146.  What  is  the  area  of  a  square  field,  of  which  the  op- 
posite corners  are  70'71  rods  apart  ?  and  what  is  the  length 
of  each  side  ?  Ans.  to  last^  50  rods,  nearly. 

147.  There  is  an"*  oblong  field,  20  rods  wide,  and  the  dis- 
tance of  the  opposite  corners  is  33^  rods  ;  what  is  the  length 
of  the  field  ?    its  area  ? 

Ans.  Length,  26§  rods;  area,  3  acres,  1  rood,  13 J  rods. 

148.  There  is  a  room  18  feet  square;  how  many  yards 
of  carpeting,  1  yard  wide,  will  be  required  to  cover  the  floor 
of  it  ?  182  _,  324  ft.  zzz  36  yards,  Ans. 

149.  If  the  floor  of  a  square  room  contain  36  square 
yards,  how  many  feet  does  it  measure  on  each  side  ? 

Ans.  18  feet 
When  one  side  of  a  square  is  given,  how  do  you  find  its 

Ofca,  or  supptficial  contents  ? 

W?ien  the  tbtea^  or  supetficial  contents,  of  a  square  is  g»V3n> 

lioiw  do  you  find  one  side  1 


tb2  MISCELLANEOUS    EXAMPLES.  TT   119, 

150.  If  an  oblong  piece  of  ground  be  SO  rods  long  and 
20  rods  wide,  wliat  is  its  area? 

A^ole,  A  Parallelogram,  or  Oblongy 
has  its  opposite  sides  equal  and  par^ 
allt'l,  but  the  adjacent  sides  unequal. 

Thus  ABC  D  is  a  parallelogram, 

£     A  F     15      and  also  E  F  C  D,  and  it  is  easy 

to  see,  that  the  contents  of  both  are 
equal.     Ans.  1600  rods,  zz:  10  acres. 

151.  What  is  the  lenr»h  of  an  oblong,  or  parallelogram, 
whose  area  is  10  acres,  and  whose  breadth  is  20  rods? 

Ans.  80  rods. 

152.  If  the  area  be  10  acres,  and  the  length  80  rods, 
what  is  the  other  side  ? 

When  the  lenr/th  and  breadth  are  given,  how  do  you  find 
the  area  of  an  oblong,  or  parallelogram  ? 

When  the  area  and  one  side  are  given,  how  do  you  find 
the  other  side  ? 

153.  If  a  board  be  18  inches  wide  at  one  end,  and  10 
inches  wide  ?.t  the  other,  what  is  the  mean  or  average  width 
of  the  board  ?  Ans.  14  inches. 

When  the  greatest  and  least  width  are  given,  how  do  you 
find  the  mean  width  ? 

154.  How  many  square  feet  in  a  board  16  feet  long,  1*8 
feet  wide  at  one  end,  and  1'3  at  the  other  ? 

Mean  width,  J-L+  ^'^  ~  1*55 ;  and  1*55  X  16  =  24*8 
feet,  Ans.  ^ 

155.  What  is  the  number  of  square  feet  in  a  board  20 
feet  long,  2  feet  wide  at  one  end,  and  running  to  a  point  at 
the  other  ?  Ans.  20  feet. 

How  do  you  find  the  contents  of  a  straight  edged  board, 
when  one  end  is  widei  than  the  other  ? 

If  the  le;igth  be  in  feet,  and  the  breadth  in  feet,  in  what 
denomination  w^iil  the  product  be  ? 

If  the  length  be  feet,  and  the  breadth  htcheSy  Vf)i2X  parts  of 
afoot  will  be  the  product? 

156.  There  is  an  obloiig  field,  40  rods  long  and  20  rods 
wide  ;  if  a  straight  line  be  drawn  from  one  corner  to  the  op- 
posite corner,  it  will  be  divided  intx)  two  equal  right-asigled 
Uiangles  ;  what  is  the  area  of  each  ? 

Ans.  400  square  rods  =  2  acres,  2  roods 


IT  119.  MISCELLANEOUS   EXAMPLES.  25S 

157.  What  is  the  area  of  a  triangle,  of  which  the  base  is 
30  rods,  and  \\\(t  perpendicular  10  rod.s?  Ans.  150  rods. 

158.  If  the  area  be  150  rods,  and  the  base  30  rods,  what 
is  the  perpendicular  ?  Ans,  10  rods. 

159.  If  the  perpendicular  be  10  rods,  and  the  area  150 
rods,  what  is  the  base  ?  Am.  30  rods. 

When  the  legs  (the  base  and  perpendicular)  of  a  right- 
angled  triangle  are  given,  how  do  you  find  its  area  ? 

When  the  area  and  one  of  the  legs  are  given,  how  do  you 
find  the  other  leg  ? 

Note,  Any  triangle  may  be  divided  into  two  right-angled 
triangles,  by  drawing  a  perpendicular  from  one  corner  to  the 
opposite  side,  as  may  be  seen  by  the  annexed  figure. 

Here  A  B  G  is  a  triangle,  di- 
vided into  two  right-angled  trian- 
gles, A  d  C,  ard  d  B  C;  there- 
fore the  whole  base^  A  B,  multi- 
plied by  one  half  the  perpendicular 
d  C,  will  givev  the  area  of  the 
whole.  If  A  B  =  60  feet,  and 
<i  C  =  16  feet,  what  is  the  area  ?  Am,  48(1  feet. 

160.  There  is  a  triangle,  each  side  of  which  is  10  feet; 
what  is  the  length  of  a  perpendicular  from  one  angle  to 
its  opposite  side  ?   and  what  is  the  area  of  the  triangle  ? 

Note.  It  is  plain,  the  perpendicular  will  divide  the  oppo- 
site side  into  two  equal  parts.     See  U  109. 

Am,  Perpendicular,  8*66  -J-  feet ;  area,  43^3    -j-  feet. 

161.  What  is  the  solid  contents  of  a  cube  measuring  6 
feet  on  each  side  ?  Ans,  216  feet. 

When  o?ie  side  of  a  cube  is  given,  how  do  you  find  its 
solid  contents  ? 

When  the  solid  contents  of  a  cube  are  given,  how  do  you 
find  one  side  of  it  ? 

162.  How  many  cubic  inches  in  a  brick  which  la  8  indies 

long,  4  inches  wide,   and  2  inches  tl^ick  ? in  2  bricks  ? 

in  10  bricks  ?  Ar»s,  to  last^  640  cubic  inches. 

163.  How  many  bricks  in  a  cubic  foot  ?  — —  in  40  cubic 
feet  ? in  1000  cubic  feet  ?  Am,  to  last^  27000. 

164.  How  many  bricks  will  it  take  to  build  a  wall  40  feet 
ia  length,  12  feet  high,  and  2  feet  thick  ?  Am,  25920. 

165.  If  a  wall  be  150  bricks,  =  100  feet,  in  length,  and 
4  bricks,  =  16  inches,  m  thijckness,  how  many  bricks  \¥ill 
lay  one  course  ?  2  courses  ? 10  courses  ?    If  Ham 

Y 


254  MISCELLANEOUS  EXAMPLES.         U  119* 

wall  be  48  courses,  nr  8  fe-et,  high,  l'to\v  many  bricks  will 
build  it  ?       150  X  4  —  600,  and  600  X  48  z='  2SS00,   Ans, 

166.  The  river  Po  is  1000  feet  broad,  and  10  feet  deep, 
and  it  runs  at  the  rate  of  4  miles  an  hour  ;  in  what  time 
will  it  discharge  a  cubic  mile  of  water  (reckoning  5000 
feet  to  the  mile)  into  the  sea?  Ans.  26  days,  1  hour. 

167.  If  the  country,  which  supplies  the  river  Po  with 
water,  be  380  miles  long,  and  120  broad,  and  the  whole 
land  upon  the  stirface  of  the  earth  be  62,700,000  square 
miles,  and  if  the  quantity  of  water  discharged  by  the  rivers 
into  the  sea  be  every  where  proportional  to  the  extent  of 
land  by  which  the  rivers  are  supplied ;  how  many  times 
greater  than  the  Po  will  the  whole  amoupt  of  the  rivers  be? 

Alls.  1375  times, 

168.  Upon  the  same  supposition,  what  quantity  of  water, 
altogether,  will  be  discharged  by  all  the  rivers  into  the  sea  in 
a  year,  or  365  days?  Aiis.   19272  cubic  miles. 

169.  If  the  proportion  of  the  sea  on  the  surlace  of  the 
earth  to  that  of  land  be  as  10^  to  5,  and  the  mean  depth  of 
the  sea  be  a  quaiter  of  a  mile  ;  how  many  years  would  it 
take,  if  the  ocean  were  empty,  to  fill  it  by  the  rivers  running 
at  the  present  rate  ?        A7is.   1708  years,  17  days,  12  hours. 

170.  If  a  cubic  foot  of  water  weigh  1000  oz.  avoirdupois, 
and  the  weight  of  mercury  be  13^  times  greater  than  of 
water,  and  the  height  of  the  mercury  in  the  barometer  (the 
weight  of  which  is  equal  to  the  weight  of  a  column  of  air 
on  the  same  base,  extending  to  the  top  of  the  atmosphere) 
be  30  inches  ;  what  will  be  the  weight  of  the  air  upon  a 

square  foot?    a   square   mile?    and  what  will  be  the 

whole  weight  of  the  atmosphere,  supposing  the  size  of  the 
earth  as  in  questions  166  and  168? 

Ans.         2109'375  lbs.  weight  on  a  square  foot. 

52734375000 mile. 

10249980468750000000 of  the  whole  atmosphere, 

171.  If  a  circle  be  14.feet  in  diameter,  what  is  its  circum- 
ference ? 

Note.  It  is  found  by  calculation,  that  the  circumference  of  a 
circle  measures  about  3.f  times  as  much  as  its  diameter,  or, 
more  accurately,  in  decimals,  3'14159  times.     Ans.  44  feet 

172.  If  a  wheel  measure  4  feet  across  from  side  to  side, 
how  many  feet  around  it  ?  Ans.   12^  feet. 

173.  If  the  diameter  of  a  circular  pond  be  147  feet,  what 
i«  its  circumfereucG  ?  Ans.  462  feet. 


t  110.  MISCELLANEOUS     EXAMPLES.  255 

174.  What  is  the  diameter  of  a  circle,  \\hose  circumfe- 
rence is  462  feet?  Ans,   147  {^tet 

175.  If  the  distance  through  the  centre  of  the  earth,  from 
side  to  side,  be  7911  miles,  how  many  miles  around  it? 

7911  X  344159  m  24853  square  miles,  nearly,  Ans. 

176.  What  is  the  area  or  contents  of  a  circle,  whose  diam- 
eter is  7  fe^^t,  and  its  circumference  22  feet  ? 

Note.  The  area  of  a  circle  may  be  found  by  multiplying 
^  the  diameter  into  4-  the  circumference,     Ans,  38^  square  feet. 

177.  Whi;t  is  the  area  of  a  circle,  whose  circumference  is 
176  rods  ?  Ans.  2464  rods. 

178.  If  a  circle  is  drawn  within  a  square,  containing  1 
square  rod,  w^hat  is  the  area  of  that  circle  ? 

Note.  The  diameter  of  the  circle  b»-ing  1  rod,  the  circum- 
ference \mII  be  344159.     Ans.  '7854  of  a  square  rod,  nearly. 

Hence,  if  we  square  the  diameter  of  any  circle,  and  multi- 
ply the  square  by  '7854,  the  product  will  be  the  area  of  the 
circle. 

179.  What  is  the  area  of  a  circle  whose  dinmeter  is  10 
rods?  102  N<  C7S54  —  78'54.  Am.  78'54  rods. 

180.  How  many  square  inches  of  leather  will  cover  a 
ball  3^  inches  in  diameter  ? 

Note.  Tlie  area  of  a  globe  or  ball  is  4  times  as  much  as 
the  area  of  a  circle  of  the  same  diameter,  and  may  be  found, 
therefore,  by  multiplying  the  whole  circumfereiice  into  the 
whole  diameter.  Ans.  384- square  inches. 

181.  What  is  the  number  of  square  miles  on  the  surface 
of  the  earth,  supposing  its  diameter  7911  miles? 

7911  X  24853  m  196,612,083,  Ans. 
1S2.  How  many  solid  inches  in  a  ball  7  inches  in  diame- 
ter? 

Note.  The  solid  contents  of  a  globe  are  f«und  by  multiply-  ' 
ing  its  area  by  ^  part  of  its  diameter. 

Am.  179f  solid  inches. 

183.  What  is  the  number  of  cubic  miles  in  the  ea.rih, 
supposing  its  diameter  as  above  ? 

Am.  259,233,0;V1,435  miles. 

184.  What  is  the  capacity,  in  cubic  inches,  of  c  hollow 
globe  20  inches  in  diameter,  and  how  much  wine  wnl  it 
contain,  1  {gallon  being  231  cubic  inches? 

Ans.  4188*8  +  cubic  inches,  and  1843  +  gallons. 

185.  There  is  a  round  log,  all  the  way  of  a  big»iess;  the 
areas  of  the   circular  ends   of  it  are  each  3  stpiare  feet; 


256  MISCELLANEOUS   EXAMPLES.  11119^. 

how  many  solid  feet  does  1  foot  in  length  of  this  log  con- 
tain ?    2  feet  in  length  ?   3  feet  ?    10  feet  ? 

A  solid  of  this  form  is  called  a  Cylinder, 

How  do  you  find  the  solid  content  of  a  cylinder,  v/hen 
the  area  of  one  endy  and  the  length  are  given  ? 

186.  What  is  the  solid  content  of  around  stick,  20  feet 
long  and  7  inches  through,  that  is,  the  ends  being  7  inches 
in  diameter  ? 

Find  the  area  of  one  end,  as  before  taught,  and  multiply  it 
by  the  length,  Ans.  6'347  ^-  cubic  feet. 

If  you  multiply  square   inches  by  inches  in  length,  what 

parts  of  a  foot  will  the  product  be  ? if  sqaare  inches  by 

feet  in  length,  what  part  ? 

187.  A  buehel  measure  is  18^5  inches  in  diameter,  and  8 
inches  deep ;  how  many  cubic  inches  does  it  contain  r 

^/i5.  2150^4  +- 

It  is  plain,  from  the  above,  that  the  solid  content  of  all 

bodies,  which  are  of  uniform  bigness  throughout,  whatever 

may  be  the  form  of  the   ends,  is  found  by  innltiplying  the 

area  of  one  end  into  its  height  or  length. 

Solids  which  decrease  gradually  from  the  base  till  they 
come  to  a  point,  are  generally  called  Pyramids.  If  the  base 
be  a  square,  it  is  called  a  square  pyramid ^  if  a  triangle,  a 
triangular  pyramid  ;  if  a  circle,  a  circular  jn/ramid,  or  a  coiie. 
The  point  at  the  top  of  a  pyramid  is  called  the  vertex,  and 
a  Jine,  drawn  from  the  vertex  perpendicular  to  the  base,  is 
called  the  perpendicular  height  of  the  pyramid. 

The  solid  content  of  any  pyramid  may  be  found  by  multi- 
plying the  area  of  the  base  by  J-  of  the  perpendicular  heights 

188.  What  is  the  lolid  content  of  a  pyramid  whose  base 
is  4  feet  square,  and  the  perpendicular  height  9  feet  ? 

42  X  I  =  48.  Ans,  48  feet. 

189.  There  is  a  cone,  whose  height  is  27  feet,  and  whose 
base  is  7  feet  in  diameter ;  what  is  its  content  ? 

Ans,  346^  feet. 

190.  There  is  a  cask,  whose  head  diameter  is  25  inches, 
hung  diameter  31  inches,  and  whose  length  is  36  inches; 

how  many  wine  gallons  does  it  contain  ?     how  many 

beer  gallons  ? 

Note.  The  mean  diameter  of  the  cask  may  be  found  by 
adding  2  thirds,  or,  if  the  staves  be  but  little  cumng,  6 
tenths,  of  the  difference  between  the  head  and  bung  diame- 


ters,  to  the  head  diameter.  The  cask  will  then  be  reduced 
to  a  cylinder. 

Now,  if  the  square  of  the  mean  diameter  be  multiplied  bj 
*7854,  (ex.  177,)  the  product  will  be  the  area  of  one  end, 
and  that,  multiplied  by  the  length,  in  inches,  will  give  the 
solid  content,  in  cubic  inches,  (ex.  185,)  which,  divided 
by  231,  (note  to  table,  wine»meas.)  will  give  the  content  in 
wine  gallons,  and,  divided  by  282,  (note  to  table,  beer  meas.) 
will  give  the  content  in  ale  or  beer  gallons. 

In  this  process  we  see,  that  the  square  of  the  mean  diame- 
ter will  be  multiplied  by  *7854,  and  divided,  for  wine  gal- 
lons, by  231.  Hence  we  may  contract  the  operation  by 
only  multiplying  by  their  quotient  {-'^%\'^  =  ^0034  ;)  that  is, 
by  '0034,  (or  by  34,  pointing  off  4  figures  from  ihe  product 
for  decimals.)  For  the  same  reason  we  may,  for  beer  gal- 
lons, multiply  by  ('-^||^  =  *0028,  nearli/y)  '0028,  &c. 

Hence  this  concise  Rule,  forguaging  or  measuring  casksy — 
Multiply  the  square  of  the  mean  diameter  by  the  length;  mtd- 
iiply  this  product  by  34  for  wine^  or  by  28  for  beer^  andj  point' 
ing  off  four  decimals^  the  product  will  be  the  content  in  gallons 
and  decimals  of  a  gallon. 

In  the  above  example,  the  bung  diameter,  31  in.  —  25  in. 
the  head  diameter  =  6  in.  difference,  and  f  of  6  =  4  inch^^s; 
25  in.  -}-  4  in,  =:  29  in.  mean  diameter. 

Then,  292  _.  S41,  and  841  X  36  in.  =  30276. 
r^.         (  30276  X  34  =:  1029384.    Ans.  102*9384  wine  gals, 
men,  ^  ,^^276  x  28  =:  847728.      Ans.    84'7728beer  gals. 

191.  How  many  wine  gallons  in  a  cask  whose  bung  diame- 
ter is  36  inches,  head  diameter  27  inches,  and  length  45 
inches?  Ans.  166*617. 

192.  There  is  a  lever  10  feet  long,  and  the  fulcrum^  or 
prop,  on  which  it  turns,  is  2  feet  from  one  end ;  how  many 
pounds  weight  at  the  end,  2  feet  from  the  prop,  will  be  bal- 
anced by  a  power  of  42  pounds  at  the  other  end,  8  feet  from 
the  prop  ? 

Note,  In  turning  around  the  prop,  the  end  of  the  lever  8 
feet  from  the  prop  will  evidently  pass  over  a  space  of  8  inches, 
wjiile  the  end  2  feet  from  the  prop  passes  over  a  space  of 
2  inches.  Now,  it  is  a  fundamental  principle  in  mechanics, 
that  the  weight  and  power  will  exactly  balance  each  othsTj 
when  they  are  inversely  as  the  spaces  they  pass  over.  Hence, 
in  this  example,  2  pounds,  8  feet  from  the  prop,  will  balance 


258  MISC£LLANEOX7S   EXAMPLES.  IT  119. 

8  pounds  2  feet  from  the  prop ;  therefore,  if  we  divide  tk€ 
distance  of  the  power  from  the  prop  by  the  distance  of  the 
WEIGHT  Jrom  the  prop^  the  quotient  will  always  express  the 
ratio  of  the  weight  to  the  power  ;  f  zz:  4,  that  is,  the  weight 
will  be  4  times  as  much  as  iht  power.    42  X  4  r=  168. 

Ans,  168  lbs. 

193.  Supposing  the  level  as tabove,  what  ^ot<?cr  would  it 
require  to  raise  1000  pounds  ?         Ans.  ^^^^  =:z  250  pounds. 

194.  If  the  weight  to  be  raised  be  5  times  as  much  as  the 
power  to  be  applied,  and  the  distance  cf  the  weight  from  the 
prop  be  4  feet,  how  far  from  the  prop  must  the  power  be 
applied  ?  Ans.  20  feet. 

195.  If  the  greater  distance  be  40  feet,  and  the  less  ^  of  a 
foot,  and  the  power  175  pounds,  what  is  the  weight? 

Anr»  14000  pounds. 

196.  Two  men  carry  a  kettle,  weighing  200  pounds  ;  the 
kettle  is  suspended  on  a  pole,  the  bale  being  2  feet  6  inches 
from  the  hands  of  one,  and  3  feet  4  inches  from  the  hands 
of  the  other  ;  how  many  pounds  does  each  bear  ? 

(  1141^  pounds- 
I    85-J-  pounds. 

197.  There  is  a  windlass,  the  wheel  of  v/hich  is  60 
inches  in  diameter,  and  the  axis,  around  which  the  rope 
coils,  is  6  inches  in  diameter;  how  many  pounds  on  the  axle 
will  be  balanced  by  240  pounds  at  the  wheel  ? 

Note.  The  spaces  passed  over  are  evidently  as  the  diame^ 
tcrSf  or  the  circumferences  ;  therefore,  ^^  zn  10,  ratio. 

Ans.  2400  pounds. 

198.  If  the  diameter  of  the  wheel  be  60  inches,  what 
must  be  the  diameter  of  the  axle,  that  the  ratio  of  the  weight 
to  the  power  may  be  10  to  1  ?  Ans.  6  inches. 

Note.  This  calculation  is  on  the  supposition,  that  there 
is  no  friction^  for  which  it  is  usual  to  add  -J  to  the  power 
which  is  to  work  the  machine. 

199.  There  is  a  screw,  whose  threads  are  1  inch  asun- 
der, which  is  turned  by  a  lever  5  feet,  =:  60  inches,  long ; 
what  is  the  ratio  of  the  weight  to  the  po^ver? 

Note,  The  power  applied  at  the  end  of  the  lever  will  de- 
scribe the  circumference  of  a  circle  60  X  2  =  120  inches 
ill  diameter,  while  the  weight  is  rais>d  1  inch ;  therefore, 
the  ratio  will  be  found  by  dividiiig  the  circumference  of  a  circle^ 
whose  diaineler  is  twice  the  length  of  the  levery  by  the  distance 


FORMS    OF    XOTES.  259 

between  the  threads  of  the  screw,     120  X  3^  =  377^  cir- 

3774. 
cumference,  and        ^  =.  377f ,  ratio,  Ajis, 

200.  There  is  a  screw,  whose  threads  are  ^  of  an  inch 
asunder;  if  it  be  turned  by  a  lever  10  feet  long,  what  weight 
will  be  balanced  by  120  pounds  power?    Ans.  30171  pounds. 

201.  There  is  a  machine,  in  which  the  power  moves  over 
10  feet,  wliile  the  weight  is  raised  1  inch  ;  what  is  the 
power  of  that  machine,  that  is,  what  is  the  ratio  of  the 
weight  to  the  power  ?  Ans,  120. 

202.  A  man  put  20  apples  into  a  wine  gallon  measure, 
which  was  afterwards  fdled  by  pouring  in  1  quart  of  water; 
required  the  contents  of  the  apples  in  cubic  inches. 

Ans.  173|-  inches. 
203  A  rough  stone  was  put  into  a  vessel,  whose  capaci- 
ty was  14  wine  quarts,  which  was  afterwards  filled  with  2^ 
quarts  of  water ;  what  was  the  cubic  content  of  the  stone  ? 

Ans,  664^  inches. 


foruss  of  sroTsa  bonds,  re- 

CBXPTS,  AMU   ORSBRS. 


NOTES. 
No.  I. 

Ovcrdean,  Sept.  17, 1802. 

For  value  received,  I  promise  to  pay  to  Oliver  Bountiful, 
or  order,  sixty-three  dollars  fifty-four  cents,  on  demand,  with 
interest  after  three  months.  William  Trustyi 

Attest  J  Timothy' Testiiviony. 

No.  II. 

Bllfort,  Sept.  17,  1802. 

For  value  received,  I  promise  to  pay  to  0.  R.,  or  bearer, 

"    "  dollars cents,  three  months  after  date. 

Peter  Pencii?* 


260  FORMS    OF    NOTES. 

No.  III. 

By  two  Persons, 

Arian,  Sept.  17, 1802. 

For  value  received,  we,  jointly  and  severally,  promise  to 

pay  to  C.  D.,  or  order, dollars  cents,  on 

demand,  with  interest.  Alden  Faithful. 

Attesty  Constance  Adley.  Jajmes  Fairface. 


Observations, 

1.  No  note  is  negotiable  unless  the  words  "  cr  order^^^  other- 
wise "  or  bcarer^^^  be  inserted  iri  it. 

2.  If  the  note  be  written  to  pay  him  "  or  order,^^  (No.  I.) 
then  Oliver  Bountiful  may  endorse  this  note,  that  is,  write 
his  name  on  the  backside,  and  sell  it  to  A,  B,  C,  or  whom 
he  pleases.  Then  A,  who  buys  the  note,  calls  on  William 
Trusty  for  payment,  and  if  he  neglects,  or  is  unable  to  pay, 
A  may  recover  it  of  the  endorser. 

3.  If  a  note  be  written  to  pay  him  "  or  bearerj^^  (No.  II.) 
then  any  person,  who  holds  the  note,  may  sue  and  recover  the 
same  of  Peter  Pencil. 

4.  The  rate  of  interest,  established  by  law,  being  six  per 
cent,  per  annum^  it  becomes  unnecessary,  in  writing  notes,  tO 
mention  the  fate  of  interest ;  it  is  sufficient  to  write  them 
for  the  payment  of  such  a  suni,  with  interest,  for  it  will  be 
understood  legal  interest,  which  is  si^l  per  cent. 

5.  All  notes  are  either  payable  oh  demand,  or  at  the  ex- 
piration of  a  certain  term  of  time  agreed  upon  by  the  parties, 
and  mentioned  in  the  note,  as  three  months,  a  year,  &c. 

6.  If  a  bond  or  note  mention  no  time  of  payment,  it  is 
always  on  demand,  whether  the  Words  "ow  demand'''^  be 
expressed  or  not, 

7.  All  notes,  payable  at  a  certain  time,  are  on  interest  a» 
soon  as  they  become  due,  though  in  such  notes  there  be  no 
mention  made  of  interest. 

This  rule  is  founded  on  the  principle,  that  every  man 
ought  to  receive  his  money  when  due,  and  that  the  non- 
payment of  it  at  that  time  is  an  injury  to  him.  The  law, 
therefore,  to  do  him  justice,  allows  him  interest  from  the 
time  the  money  becomes  due,  as  a  cou.pensation  for  the 
injury. 

8.  Upon  the  same  principle,  a  note,  payao'*^  on  demand, 
without  any  mention  made  of  interest,  is  on  inioicst  after  » 


f'OKMS    OF    BONDS.  261 

demand  of  payment,  for  upon  demand  snch  notes  imme- 
diately become  due. 

9.  If  a  note  be  given  for  a  specific  article,  as  rye,  payable 
in  one,  two,  or  three  months,  or  in  any  certain  time,  and  the 
signer  of  such  note  sutfers  the  time  to  ehipse  without  de- 
livering such  article,  the  holder  of  the  note  w411  not  be 
obliged  to  take  the  article  afterwards,  but  may  demand  and 
recover  the  value  of  it  in  money. 


BONDS. 

A  Bond,  with  a  Condition,  from  one  to  another. 

Know  all  men  by  these  presents,  that  I,  C.  D.  of,  &c.,  in 
the  county  of,  &c.,  am  held  and  firmly  bound  to  E.  F.,  of, 
&c.,  in  two  hundred  dollars,  to  be  paid  to  the  said  E.  F.,  or 
his  certain  attorney,  his  executors,  administrators,  or  assigns, 
to  which  payment,  well  and  truly  to  be  made,  I  bind  myself, 
my  heirs,  executors  and  administrators,  firmly  by  these 
presents.  Sealed  with  my  seal.  Dated  the  eleventh  day  of 
,  in  the  year  of  our  Lord  one  thousand  eight  hun- 
dred and  two. 

The  Condition  of  this  obligation  is  such,  that,  if  the  above- 
bound  C.  D.,  his  heirs,  executors,  or  administrators,  do  and 
shall  well  and  truly  pay,  or  cause  to  be  paid,  unto  the  above- 
named  E.  F.,  his  executors,  administrators,  or  assigns,  the 
full  sum  of  two  hundred  dollars,  with  legal  interest  for 
the  same,  on  or  before  the  eleventh  day  of next  en- 
suing the  date  hereof, — then  this  obligation  to  be  void,  or 
otherwise  to  remain  in  full  force  and  virtue. 

Signed,  &c. 


A  Condition  of  a  Counter  Bond,  or  Bond  of  Indemnity,  where 
one  man  becomes  hound  for  another. 
The  condition  of  this  obligation  is  such,  that  whereas  the 
above-named  A.  B.,  at  the  special  instance  and  lequest,  and 
for  the  only  proper  debt  of  the  above-bound  C.  D.,  together 
with  the  said  C.  D.,  is,  and  by  one  bond  or  obligation  bear- 
ing equal  date  with  the  obligation  above-written,  held  and 

firmly  bound  unto  E.  F.,  of,  &:c.,  in  the  penal  sum  of 

dollars,  conditioned  for  the  payment  of  the  sum  of,  &c.,  with 
legal  interest  for  the  same,  on  the day  of         ■ '  ■ 


862  J'ORMS    OF    RECEIPTS. 

next  ensuing  the  date  of  the  said  in  part  recited  obligation, 
as  in  and  by  the  said  in  part  recited  bond,  with  the  conditiop 
thereunder  written,  may  more  fully  appear ; — if,  therefore,  the 
said  C.  D.,  his  heirs,  executors,  or  administrators,  do  and 
shall  well  and  truly  pay,  or  cause  to  be  paid,  unto  the  said 
E.  F.,  his  executors,  admini.'strators,  or  assigns,  the  said  sum 

of,  6ic.,  with  legal  interest  of  the  same,  on  the  said 

day  of,  &c.,  next  ensuing  the  date  of  the  said  in  part  re- 
cited obligation,  accordirg  to  the  true  intent  and  meaning, 
and  in  full  discharge  and  satisfaction  of  the  said  in  part  re- 
cited bond  or  obligation, — then,  &c. — otherwise,  &c. 


Note,  The  principal  difference  between  a  note  and  a 
bond  is,  that  the  latter  is  an  instrument  of  more  solemnity, 
being  given  under  seal.  Also,  a  note  may  be  controlled  by 
a  special  agreement,  different  from  the  note,  whereas,  in  case 
of  a  bond,  no  special  agreement  can  in  the  least  control 
what  appears  to  have  been  the  intention  of  the  parties,  as» 
expressed  by  the  words  in  the  condition  of  the  bond. 

-iversity)    receipts. 

^/     -  rr-^^,^  Sitgrieves,  Sept.  19, 1802. 

Received  from  Mr.  Durance  Adley  ten  dollars  in  full 
of  aH  accounts.  Orvand  Constance. 


^' 


Sit^ieves,  Sept.  19, 1802. 

Received  of  Mr.  Oiivand  Constance  five  dollars  in  full 
of  all  accounts.  Durance  Adlev. 


Receipt  for  Money  received  on  a  Note. 

Sitgr-eves,  Sept.  19,  1802. 

Received  of  Mr,  Simpson  Eastly  (by  the  hand  of  Titus 
Trusty)  sixteen  dollars  twenty-five  cents,  which  is  en- 
dorsed on  his  note  of  June  3,  1S02. 

Peter  Cheerful. 


A  Receipt  for  Money  received  on  Acco'unt^ 

Sitgricves,  Sept.  19, 1802. 

Received  of  Mr.  Grand  Landike  fifty  dollars   on   ac- 
count Ei PRO  Slackley, 


FORMS    OF  RECEIPTS,  &C. BOOK-KEEPING.         203 

Receipt  for  Money  received  for  another  Person. 

Salem,  Aug.  10, 1827. 

Received  from  P.  C.  oae  hundred  dxjilars  for  account  of 
J.  B.  Eli  Tuuman. 


Receipt  for  Interest  due  on  a  Note. 

Amherst,  July  6, 1827. 
Received  of  I.  S.  thirty  dollars,  in  full  of  one  year's  in- 
terest of  $  500,  due  to  me  on  the day  of  • 


last,  on  note  from  the  said  I.  S.  Solomon  Gray. 

Receipt  for  Money  paid  before  it  becomes  due. 

Hillsborough,  May  3, 1827. 

Received  of  T.  Z.  ninety  dollars,  advanced  in  full  for  one 
year's  rent  of  my  farm,  leased  to  the  said  T.  Z.,  ending  the 
first  day  of  April  next,  1828.  Honestus  James. 

Note.  There  is  a  distinction  between  receipts  given  in 
full  of  all  accounts^  and  others  in  full  of  all  demands.  The 
former  cut  off  accounts  only  ;  the  latter  cut  off  not  only  ac- 
counts, but  all  obligations  and  right  of  action. 


ORDERS. 

Archdal*?,  Sept.  9, 1802. 
Mr.  Stephen  Burgess.     For  value  received,  pay  to  A* 
B.,  or  order,  ten  dollars,  and  place  the  same  to  my  account 

Samuel  Skinner« 


Pittsburgh,  Sept.  9,  1821. 

Mr.  James  Rorottom.  Please  to  deliver  Mr.  L.  D.  such 
goods  as  he  may  call  for,  not  exceeding  the  sum  of  twenty- 
five  dollars,  and  place  the  same  to  the  account  of  your 
humble  servant,  Nicholas  Reubens. 


It  is  neceesary  that  every  man  should  have  some  regular, 
uniform  method  of  keeping  his  accounts.  What  this  method 
ghall  be,  the  law  does  not  prescribe ;  but,  in  cases  of  dis- 
pute, it  requires  that  the  book,  or  that  on  which  the  charges 
were  originally  made,  be  produced  in  open  court,  when  b^ 
will  be  required  to  answer  to  the  following  questiong :— 


m4 


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